Chapter Summaries For University Physics

Chapter 1Physical World and Measurement

TARGET CBSE

1. Science is a study of the physical and natural world using theoretical models and data from experiments, observa-tions, and predictions.

2. Theory refers to a description of the world that covers a relatively large number of phenomena and has met and explained many observational and experimental tests.

3. The word Physics originated from the Greek word phusis, which means nature, introduced by Aristotle. Physics is the branch of science which is devoted to the study of nature and natural phenomena based on the fact that all the events in nature take place according to some basic laws. Physics reveals these basic laws from day-to-day observa-tions.

4. A hypothesis is a supposition without assuming that it is true, an axiom is a self-evident truth, and a model is a theory proposed to explain the observed phenomena.

5. Four types of forces that exist in nature are gravitational force, electromagnetic force, strong nuclear force, and weak nuclear force.

6. Gravitational force is inversely proportional to the square of the distance between the two bodies and directly proportional to the product of the masses of two bodies.

7. Physical quantities that stay unchanged in a process are considered as conserved quantities. In nature, we have some conservation laws, namely, laws of conservation of mass, energy, linear momentum, angular momentum, charge, parity, and so on. Some conservation laws are true for one fundamental force but not for the other. Conser-vation laws are connected with symmetries of nature. Symmetries of space and time, and other types of sym-metries, play a major role in modern theories of funda-mental forces in nature.

8. Physical quantity is a quantity that can be measured and by which various physical happenings can be explained and expressed in form of laws.

9. There are three fundamental physical quantities (length, mass, and time), which are the physical quantities that do not depend on other physical quantities.

10. Derived physical quantities are the physical quantities that can be derived by suitable multiplication or division of different powers of fundamental quantities (e.g., area, volume, and force).

11. Unit is a unique name we assign to a measure of a physical quantity (such as length, time, mass, temperature, pressure, and electric current). For example, meter (m) is the measure or unit of the physical quantity “length.”

12. Any unit that cannot be expressed in terms of other units are called fundamental units. Any unit of mass, length, or time in mechanics is called a fundamental, absolute, or base unit. Other units, which can be expressed in terms of funda-mental units, are called derived units.

13. A complete set of units, both fundamental and derived for all kinds of physical quantities, is called system of units. The common systems are as follows: CGS system – centi meter (cm), gram (g), and second (s); MKS system – meter, kilogram, and second; FPS system – foot, pound, and second are used for measurements of length, mass, and time, respectively. In this system, force is a derived quantity with unit poundal; SI system – It is known as International System of Units, and is in fact an extended system of units applicable to whole physics.

14. The standard for a unit has the following features: It should be well defined; It should be invariable (should not change with time); It should be convenient to use; It should be easily accessible.

15. The 14th General Conference on Weights and Measures (held in France) picked seven quantities (length, mass, time, temperature, electric current, luminous intensity, amount of substance) as base quantities, thereby form-ing the International System of Units abbreviated as SI (System de International) system.

16. One meter (1 m) is the length of the path traveled by light in a vacuum during a time interval of 1/299, 792, 458 of a second.

17. One second (1 s) is the time taken by 9, 192, 631, 770 oscillations of the light (of a specified wavelength) emit-ted by a cesium-133 atom.

18. Unit of speed = (Unit of distance (length))/(Unit of time) = m/s = ms–1 (should be read as meter per second).

19. The international 1 kg standard of mass is a platinum–iridium cylinder 3.9 cm in height and in diameter.

20. If equal currents are maintained in the two wires so that the force between them is 2 × 10–7 newton per meter of the wires, the current in any of the wires is called 1 A. Here, newton is the SI unit of force.

21. Numbers expressed with the aid of powers of 10 are said to be in scientific notation.

22. The change in position of the pencil when viewed from two different eyes is known as parallax. Parallax method is used to determine distances of nearby stars.

23. One-twelfth of the mass of an atom of carbon–12 isotope (12C) including the mass of electrons (1.67 × 10–27 kg) is known as unified atomic mass unit.

MTPL0139 Summary Ch-1.indd 1 4/10/2012 8:55:41 PM

Chapter 1 Physical World and Measurement2

24. Mass spectrograph is a device used to measure the mass of atomic/sub-atomic particles such as electron and isotopes of element.

25. Accuracy of a measurement means that how close the measured value is with the true value of physical quantity. The limit or resolution to which a physical quantity is meas-ured by a measuring instrument is known as precision.

26. The uncertainty in measurement of an instrument is known as error. In turn, every calculated physical quan-tity (derived physical quantity), which depends upon the measured fundamental quantity, will also contain error.

27. Errors that are unidirectional in nature, which means they are either positive or negative, are known as systematic errors. For example, personal error, instrumental errors, and errors due to imperfection in experimental technique or procedure.

28. The errors occurring due to unpredictable variations in experimental conditions such as unpredictable fluctuations in temperature, voltage supply, and mechanical vibrations of experimental setups are known as random errors.

29. The smallest value that can be measured by the meas-uring instrument is the least count of the instrument. Smaller the least count of a measuring device, higher is the accuracy of measurement.

30. Absolute error in the measurement of a physical quantity is the magnitude of the difference between the true value and the measured value of the quantity.

31. Mean absolute error is the arithmetic mean of the magnitudes of absolute errors in all the measurements of the quantity.

32. The relative error or fractional error of measurement is defined as the ratio of mean absolute error to the mean value of the quantity measured.

33. When the relative/fractional error is expressed in percentage, we call it percentage error.

34. Significant figures are the measured values of physical quantities that tell the number of digits in which we have confidence.

35. Rounding off a decimal is a technique used to estimate or approximate values. Rounding is generally used to limit the amount of decimal places; instead of having a long string of decimals places, we can approximate the value of the decimal to a specified decimal place.

36. The result of an addition or subtraction in the number having different precisions should be reported to the same number of decimal places as are present in the number having the least number of decimal places.

37. The answer to a multiplication or division is rounded off to the same number of significant figures as is possessed by the least precise term used in the calculation.

38. Dimensions are the powers to which fundamental quan-tities must be raised in order to express the given physical quantity.

39. To express dimensions, each fundamental unit is repre-sented by a capital letter (e.g., the unit of length is denoted by L, unit of mass by M, unit of time by T, unit of electric current by I, unit of temperature by K, and unit of lumi-nous intensity by C ).

40. According to the principle of homogeneity, we can add and multiply physical quantities with same or different dimensional formulas at our convenience, however, no such rule applies to addition and subtraction, where only like physical quantities (scalars, and vectors with vectors) can only be added or subtracted.

TARGET COMPETITION

1. Physics covers a wide range of phenomena – from the smallest subatomic particles to the largest galaxies. Included in this are the very most basic objects from which all other things are composed of and therefore physics is sometimes said to be the fundamental science.

2. Gravitational force is always attractive in nature.

3. Electromagnetic force can be either attractive or repulsive depending upon the type of charges.

4. The strong nuclear force is the strongest force among the four fundamental forces of nature, and it is always attractive.

5. It is not necessary that a dimensionally correct equation is also physically correct but a physically correct equation has to be dimensionally correct.

6. The powers of 10 and the zeros on the left side of the measurement are not counted while counting the signifi-cant figures.

7. Greater the number of significant figures in a measure-ment, smaller is the percentage error.

8. For rounding off significant figures, if the succeeding figure is greater than 5, the figure is increased by 1 or else it is left unchanged. However, if the succeeding figure is 5 itself, the figure is raised by 1 if it is odd and left unchanged it is even.

9. The error in a measurement is always equal to the least count of the measuring instrument.

10. Constants never contribute in the case of errors.


Chapter 2Motion in a Straight Line

TARGET CBSE

1. A matter having a finite size, shape, and occupying some definite space is called a body.

2. Change in position of an object with time is known as motion.

3. An object is said to be at rest if its position does not change with time with respect to a reference point that is generally stationary or with respect to its surrounding objects.

4. Motion of a body can be of three types: (1) rectilinear or translatory motion, (2) circular or rotatory motion, and (3) oscillatory or vibratory motion.

5. Rectilinear motion is that motion in which a particle or point mass body moves along a straight line.

6. If a particle or point mass body moves on a circle, such a motion is called circular motion.

7. If an object is small enough so that we can treat it similar to a point – we call such an object a point object or point particle.

8. A frame of reference is a set of coordinate axes in terms of which position of an object with reference to which physical laws may be mathematically stated.

9. Position vector describes the instantaneous position of a particle with respect to the chosen frame of reference.

10. Distance is a measure of the length of the path taken during the change in position of an object. Distance is a scalar quantity. It does not specify a direction.

11. Displacement is a measure of the change in position of an object. Displacement is a vector quantity.

12. The rate of change of position with time is known as velocity. It is a vector quantity whose magnitude is simply the speed of the particle.

13. The average velocity of a moving particle over a certain time interval is defined as the displacement per unit time.

14. Average speed which is a scalar quantity is expressed as

Average speed = DistanceTime interval

.

15. The velocity at a particular instant of time is known as instantaneous velocity.

16. Acceleration is the rate of change of velocity with time for a moving particle.

17. The average acceleration for a finite time interval is defined as average acceleration, which is written as

Average acceleration = Change in velocityTime interval

.

18. Average acceleration is a vector quantity whose direction is same as that of the change in velocity.

avtavg = ∆

∆

.

19. The acceleration at a given instant of time is the instanta-neous acceleration.

20. If the speed of an object remains constant, then the motion is called uniform motion and if speed is a vari-able, then motion is called non-uniform motion.

21. The position–time graph for uniform motion is shown in Fig. 2.1.

x2 = x1 + v(T - t),

where v is the slope of the graph which is constant.

Position

x2

Time

TimeT

x1

Figure 2.1 22. instantaneous velocity is defined as the rate of change

of displacement.

23. The velocity–time graph for non-uniform motion is shown in Fig. 2.2.

Velocity

v1

v2

t2 Timet1

Figure 2.2 24. Distance traveled in nth second is given by

x va

nn = + -0 22 1( ),

where v0 is the initial velocity, a is the uniform acceleration, xn is the distance traveled in nth second.

25. The velocity–time graph (Fig. 2.3) gives three types of information: (1) The instantaneous velocity; (2) the slope of the tangent to the curve at any point gives instantaneous acceleration a = dv/dt = tan q; (3) the area under the curve gives total displacement of the particle:

x v dtt

t

= ∫ .1

2


Chapter 2 Motion in a Straight Line4

v2

v

v1

t2 tt1

Displacement

Figure 2.3

26. Acceleration time curves (Fig. 2.4) tell us about the varia-tion of acceleration with time. Area under the acceleration–time curve gives the change in velocity of the particle in the given time interval.

t

t

t

a a a

Motion under uniformacceleration

Motion under uniformretardation

Motion under variableacceleration

Figure 2.4

27. When a body is dropped from some height, it falls freely under gravity with constant acceleration g (= 9.8 m/s2) provided the air resistance is negligible small.

28. For freely falling object, the three equations of motion are (1) v = v0 + at, (2) v2 = v0

2 + 2ax, and (3) x = v0t + (1/2)at2, where x is the displacement of the body and v0 is initial velocity of projection in the vertically downward direc-tion. However, if an object is projected vertically upward with initial velocity v0 , the equations of motion are (1) v = v0 - at, (2) v2 = v0

2 - 2ax, and (3) x = v0t - (1/2)at2.

29. The velocity with which an object moves with respect to another object is known as relative velocity of that object.

TARGET COMPETITION

1. Kinematics deals with the concepts that are needed to describe motion without any reference to forces. Dynamics deals with the effect that forces have on motion. Kinemat-ics and dynamics together form the branch of physics which is known as mechanics.

2. When the position of object changes on a straight line, that is, when the motion of an object is a along straight line, the motion is called one-dimensional motion.

3. For a body in motion displacement can be zero but distance (or path length) can never be zero.

4. For a straight line motion, if a particle goes from point A to point B, the displacement and distance are equal.

5. Distance can never be zero, but displacement can be zero.

6. If a particle goes from A to B along a curve, the distance is the length of the total path covered by that particle whereas displacement is the shortest distance between points A and B.

7. The average velocity is a vector in the direction of dis-placement. It depends only on the net displacement and the time interval.

8. A large magnitude of acceleration indicates that the veloc-ity is changing very rapidly.

9. For object in motion, the position–time graph (Fig. 2.5) is a straight line parallel to time axis. Slope of this graph is a straight line.

tTime

Posi

tion

x

0

Figure 2.5

10. For an object moving with constant speed, the position–time graph is a straight line (Fig. 2.6) inclined to time axis.

x

∆x∆x

∆tq

∆t tTime

Posi

tion

Figure 2.6 11. For non-uniform motion, position–time graph is a curve

(Fig. 2.7).

x

(Pos

itio

n)

A

O t1 t2 t3 t(Time)

BC

D

Figure 2.7

12. Acceleration–time graph for uniformly accelerated motion is as shown in Fig. 2.8.

a

Acc

eler

atio

n

a0

tTime

Figure 2.8

13. Velocity–time graph for uniformly accelerated motion with initial velocity v0 and initial position x0 are shown in Fig. 2.9.


Target Competition 5

v v v

t t t

If v0= 0

Velo

city

Velo

city

Velo

city

Time Time TimeIf v0 > 0

qq

q

If v0 < 0

Figure 2.9

14. Position–time graph for uniformly accelerated motion with initial velocity v0 and initial position x0 are shown in Fig. 2.10.

t t t

x

x0

x x

x0x0

If v0= 0

Posi

tion

Posi

tion

Posi

tion

Time Time TimeIf v0 > 0 If v0 < 0

Figure 2.10

15. Acceleration–time graph for uniformly retarded motion is shown in Fig. 2.11.

a

t

TimeAcc

eler

atio

n

−a0

Figure 2.11

16. Velocity–time graph for uniformly retarded motion with initial velocity v0 and initial position x0 are shown in Fig. 2.12.

v v

t t

v

t

If v0= 0

Time Time TimeVelo

city

Velo

city

Velo

city

If v0 > 0

q q q

If v0 < 0

Figure 2.12

17. Position–time graphs for uniformly retarded motions with initial velocity v0 and initial position x0 are shown in Fig. 2.13.

x

t t t

x0

x

x0

x

x0

If v0= 0

Posi

tion

Posi

tion

Posi

tion

Time Time TimeIf v0 > 0 If v0 < 0

Figure 2.13

18. If two objects are moving in same direction along straight line, the magnitude of relative velocity of one object with respect to the other object will be equal to difference in magnitude of the velocities of the two objects.

19. If two objects are moving in opposite direction along straight line, the magnitude of relative velocity of one object with respect to the other object will be equal to sum of the magnitude of the velocities of the two objects.


Chapter 3Motion in a Plane

TARGET CBSE

1. Galileo proposed what we now call the law of compound motion, according to which the motion in one dimension has no effect on motion in another dimension.

2. Those physical quantities which have both magnitude as well as direction are known as vector quantities (e.g., dis-placement, velocity, acceleration, force, etc.)

3. Physical quantities such as temperature, pressure, energy, mass, and time that do not involve direction are called scalar quantities. A single value with a sign (as in a tem-perature of − 40° C) specifies a scalar.

4. A vector with magnitude zero and an arbitrary direction is called a null vector.

5. Types of vectors:

(a) A vector is said to be a negative vector of another vector if their magnitudes are equal but directions are opposite.

(b) Two vectors are said to be equal vectors if they have same magnitude and direction.

(c) Two vectors are said to be unequal vectors if they have different magnitudes or different directions or both.

(d) Two vectors are said to be parallel vectors if they have the same direction. Their magnitudes may or may not be equal. All equal vectors are parallel but the reverse may not be true.

(e) Two vectors are said to be anti-parallel vectors if they point in opposite directions. Their magnitude may or may not be equal.

(f) Two or more vectors which are in the same line or can be brought in the same line by shifting them parallel to each other are known as collinear vectors.

(g) Those vectors which lie in the same plane are called coplanar vectors.

(h) If two or three vectors are perpendicular to each other they are known as orthogonal vectors. Best example of orthogonal vectors is the Cartesian coordinate axes.

(i) A vector having magnitude equal to unity but having a specific direction is called a unit vector.

(j) Vectors which have common initial point are known as co-initial vectors.

(k) Vectors whose initial point is fixed they are known as localized vectors. For example, position vector is a localized vector because its initial point always lies at the origin.

(l) Vectors whose initial point is not fixed are known as non-localized vectors.

(m) A vector which gives the position of a point with reference to the origin of the coordinate system is called the position vector of the point.

6. The position vector

r at any time t, in terms of two- dimensional coordinates x and y is given by

r x y= + or

r xi y j= + ,

where the magnitude

r r x y= = +2 2 .

7. The position vector

r at any time t, in terms of three- dimensional coordinates x, y, and z is given by

r x y z= + + or

r xi y j zk= + + .

where the magnitude

r r x y z= = + +2 2 2 .

8. Let us consider three vectors

A B C, , and , as shown in Fig. 3.1. They are equal if the direction and magnitude of all the three vectors is the same; that is, if

A B C= = and A = B = C

A

B

C

Figure 3.1

If we shift vectors

A B, , and

C parallel to themselves all the three vectors will coincide with each other, which means that tails and tips of

A B, , and

C will coincide with each other.

9. Vector addition: Vector

s is the vector sum of vectors

a and

b :

s a b= + .

10. Commutative law:

a b b a+ = + .

11. Associative law: ( ) ( ).

a b c a b c+ + = + +

12. Vector subtraction:

d a b a b= − = + −( ).

13. If we know a vector in component notation (ax and ay) and if we want it in magnitude – angle notation (a and q), to transform it, we can use the following equations:

a a aa

ax yy

x

= + =2 2 and tan .θ

14. In the more general three-dimensional case, we need a magnitude and two angles (say, a, q, and f) or three com-ponents (ax, ay, and az) to specify a vector.

15. unit vectors in the positive directions of the x, y, and z axes are labeled i j , and k , respectively, where the hat or cap symbol ^ is used instead of an overhead arrow as for other vectors. Unit vectors are very useful for expressing other vectors:


Chapter 3 Motion in a Plane8

a a i a jx y= + ;

b b i b jx y= + ,

where the quantities a ix and a jy

are vectors – called the vector components of

a. The quantities ax and ay are scalars – called the scalar components of

a.

16. A third way to add vectors is to combine their components axis by axis:

dx = ax – bx, dy = ay – by and dz = az – bz

where ˘ .z d d i d j d kx y z= + +

17. There are three laws for the addition of vectors: (1) Triangle law of vectors for addition of two vectors; (2) Parallelogram law of vectors for addition of two vectors; (3) Polygon law of vectors for addition of more than two vectors.

18. Triangle law of vectors: If two vectors can be represented both in magnitude and direction by the two sides of a tri-angle taken in the same order, then the resultant is repre-sented completely, both in magnitude and direction, by the third side of the triangle taken in the opposite order.

19. It follows from triangle law of vectors that if three vectors

A,

B , and

C and can be represented completely by the three sides of a triangle taken in order, then their vector sum is zero:

A B C+ + = 0.

20. Parallelogram law of vectors: When two vectors

P and

Q are completely represented by the two sides OA and OB of a parallelogram, then, according to the parallelogram law of vectors, the diagonal OC of the parallelogram will be the resultant

R , such that

R P Q= + .

21. Polygon law of vectors: If a number of vectors can be represented both in magnitude and direction by the sides of an open convex polygon taken in the same order, the resultant is represented completely in magnitude and direction by the closing side of the polygon, taken in the opposite order. If a number of vectors are represented by the sides of a closed polygon taken in order, their resultant is zero.

22. An equilibrant vector is a single vector which balances two or more vectors acting simultaneously at a point.

23. When a particle moves, the position vector changes – say, from

r1 to

r2 during a certain time interval – then the particle’s displacement Δ

r during that time interval is

Δ

r r= 2 1− r .

24. If a particle moves through a displacement Δ

r in a time interval Δt, its average velocity

vavg is expressed as follows:

Average velocity = DisplacementTime interval avg⇒ =

vrt

ΔΔ

.

25. When a particle’s velocity changes from

v1 to

v2 in a time interval Δt, its average acceleration

aavg during the time interval Δt is expressed as follows:

Average acceleration = Change in velocity

Time interval⇒

aavvg =−

=

v v

t

v

t2 1Δ

Δ

Δ.

26. The velocity is otherwise called as instantaneous velocity, which is given by the limiting value of the average velocity as the time interval approaches zero:

vt

d rdtt

= =→

lim .Δ

ΔΔ0

r

27. A particle moves in a vertical plane with some initial velocity

v0 but its acceleration is always the free-fall accel-eration

g, which is downward – such a particle is called a projectile (meaning that it is projected or launched) and its motion is called a projectile motion. The path followed by a projectile during its motion is known as its trajectory.

28. In projectile motion, the horizontal motion and the verti-cal motion are independent of each other, that is, neither motion affects the other.

29. The horizontal distance covered by a projectile from its original or initial position (x = y = 0) to the position where it passes y = 0 during its fall is called the horizontal range (R).

30. The total time taken by the projectile to return to the same level from where it was thrown is known as time of flight.

31. There are two angles of projection for the same horizontal range, that is, q and (90° – q). The projectile will cover the same horizontal range whether it is thrown at an angle (90° – q) with the horizontal, or an angle q with the vertical.

32. A particle that travels around a circle or a circular arc at constant (uniform) speed is said to be undergoing uniform circular motion.

33. Centripetal is an adjective that describes any force or superposition of forces that is directed toward the center of curvature of the path of motion.

34. A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the body’s speed.

35. An acceleration that is directed radially inward is called a centripetal acceleration.

36. In a circular motion, the distance traveled by a particle in one revolution is just the circumference of the circle (2pr). The time for a particle to go around a closed path exactly once has a special name – the period of revolution or simply the period of the motion. The period is repre-sented with the symbol T, which is expressed as

Tr

v= 2π

.

37. In a circular motion, the total number of revolutions by a particle in a given time is known as the frequency (f ) of revolution. From the definitions we have given for period and frequency, they are related by the expression

fT

= 1.



1. One general way of locating a particle (or particle-like object) is with a position vector r , which is a vector that extends from a reference point (usually the origin). In terms of Cartesian coordinates,

r can be written as

r xi y j zk= + + ,

where xi y j zk

, ,and are the vector components of

r and the coefficients x, y, and z are its scalar components.

2. When a particle moves, the position vector changes – say, from

r1 to

r2 during a certain time interval – then the par-ticle’s displacement Δ r during that time interval is

Δ = −

r r r2 1.

3. If a particle moves through a displacement Δ

r in a time interval Δt, its average velocity

vavg is

vrtavg = Δ

Δ.

The direction of

vavg must be the same as that of the dis-placement Δ

r .

4. A moving particle’s instantaneous velocity,

v , is defined as the derivative

vdrdt

= .

5. The direction of the instantaneous velocity

v , of a particle is always tangent (Fig. 3.2) to the particle’s path at the par-ticle’s position.

r1r2

Path

Tangent

O

y

x

12

r∆

Figure. 3.2

6. When a particle’s velocity changes from

v1 to

v2 , in a time interval Δt, its average acceleration

aavg , during Δt is

av v

tvtavg = − =2 1

ΔΔΔ

.

7. If Δt→ 0, then in the limit

aavg , the particle approaches the instantaneous acceleration

advdt

= .

8. If the velocity changes in either magnitude or direction (or both), the particle must have acceleration. We can rewrite acceleration

a as

a a i a j a k

dvdt

idv

dtj

dvdt

k

x y z

x y z

= + +

= + + .

9. To find the scalar components of

a, we differentiate the scalar components of

v . Figure 3.3 shows an acceleration vector

a and its scalar components for a particle moving in two dimensions.

O

y

x

ay

ax

Path

a

Figure 3.3

10. A particle moves in a vertical plane with some initial velocity

v0 , but its acceleration is always the free-fall acceleration g, which is downward. Such a particle is called a projectile and its motion is called projectile motion. We can analyze projectile motion using the tools for two-dimensional motion assuming that air has no effect on the projectile. The projectile is launched with an initial velocity

v0 that can be written as

v v i v jx y0 0 0= + .

The components v0x and v0y can then be found if we know the angle q0 between v0 and the positive x-direction:

v v v vx y0 0 0 0 0 0= =cos and sin .θ θ

11. In projectile motion, the horizontal motion and the vertical motion are independent of each other, that is, neither motion affects the other.

12. Horizontal motion: Because there is no acceleration in the horizontal direction, the horizontal component vx of the projectile’s velocity remains unchanged from its initial value v0x throughout the motion. At any time t, the projec-tile’s horizontal displacement x – x0 from an initial position x0 with a = 0 is given by

x x v tx− =0 0 .

Because v vx0 0 0= cos ,θ this becomes

x x v t− =0 0 0( cos ) .θ

13. Vertical motion: In a vertical motion the acceleration is constant. Thus, the earlier discussed constant acceleration equations apply, provided we substitute – g for a and switch to y-notation. Then, we can rewrite the equations of motion as

TArgeT COMPeTiTiON


Chapter 3 Motion in a Plane10

y y v t gt

v t gt

y− = −

= −

0

0

02

02

12

12

( sin ) .θ

Therefore,

v v gty = −0 0sin .θ

v v g y yy

20 0

2 2= − −( sin ) ( ).θ 0

The vertical velocity component is directed upward ini-tially, and its magnitude steadily decreases to zero, which marks the maximum height of the path. It then reverses direction, and its magnitude becomes larger with time.

14. equation of path: The equation of the projectile’s path (its trajectory) is given by

y xgx

v= −(tan )

( cos ).θ

θ0

2

2 0 0

15. The horizontal range R of the projectile is the horizontal distance the projectile has traveled when it returns to its initial (launch) height, which is given by

Rvg

= 02

02sin .θ

The horizontal range R is maximum for a launch angle of 45°.

16. The equation of a trajectory in standard form can be mod-ified and expressed in terms of R as

y xxR

= −

tan .θ 1

17. Air resistance: Generally it is assumed that the air through which the projectile moves has no effect on its motion. However, in many situations, the disagreement between general calculations and the actual motion of the projectile can be large because the air resists (opposes) the motion.

18. Motion in inclined plane: Suppose there is an inclined plane making an angle q with the horizontal and a particle is projected with speed v0 at an angle a from the horizon-tal (Fig. 3.4). Now we have to take x-axis along the inclined plane and y as line perpendicular to x.

H

a q R

y

v0

x

Figure 3.4

Along x axis Along y axis Initial velocity: Initial velocity: v vx0 = −0 cos( )α θ v vy0 = −0 sin( )α θ Acceleration: Acceleration: a gx = − sinθ a gy = − cosθ Velocity at any time t: Velocity at any time t: v v g tx = − −0 cos( ) sinα θ θ v v g ty = − −0 cos( ) cosα θ θ It is still same as the original projectile motion with con-

stant acceleration.

19. The time in which the projected particle strikes the inclined plane is called time of flight. Let the time of flight be given by T. Since displacement in y direction is y – y0 = 0, from the constant acceleration equation, we can write

012

2= − −v T g T0 sin( ) cosα θ θ

Tv

g= −2 0 sin( )

cos.

α θθ

20. It is the maximum height to which the projected particle rises relative to the inclined plane. Let the maximum height be given by H. The maximum height relative to the inclined plane is attained where vy = 0. Hence, we can write

( ) sin ( ) cos .0 2202 2= − −v g Hα θ θ

Therefore,

Hv

g= −0

2

2sin ( )

cos.

α θθ

21. When an object moves in a circular path at a constant speed, the motion of the object is called uniform circular motion.

22. Angular velocity is defined the rate of change of angle swept by the radius by a particle that moves in circular motion:

w = dq/dt,

where angular velocity is expressed in radians per second (rad/s).

23. For a particle moving in circular motion, velocity is directly proportional to radius for a given angular velocity

v = w R.

24. For uniform circular motion (or for a motion with constant angular velocity), the motion is periodic, which means particle passes through each point of the circle at equal intervals of time. Time period of motion is given by

T = 2p / w .

25. Angular acceleration is defined as the rate of change of angular velocity moving in circular motion with time:

a = dw / dt = d2q / dt2,

where angular acceleration is measured in rad/s2.



26. The angular position at any time t is given by

q = w t + (1/2)a t2.

27. Motion in two dimensions: In a two-dimensional situa-tion, consider three observers at rest O, A, and B. Position of B from reference frame of A is

rBA . For A, the origin of reference frame is not O, it is A itself. From the law of vec-tors we can deduce that

r r r r r rB A BA BA B A; .= + = −thus,

Position vector of B with respect to A is defined as

rBA . Differentiating this equation with respect to time, we get

v v vΒΑ = −B A .

Differentiating further, we get

a a aBA B A .= −

28. Motion in three dimensions: In three dimensions, three units vectors i j k

, , and associated with each coordinate axis of Cartesian coordinates system exist as shown in Fig. 3.5. Consider a particle moving in three-dimensional space. Let P be its position at any point t. The position vec-tor of this particle at point P is

r xi y j zk= + + .

where x, y, and z are coordinates of point P. Similarly, velocity and acceleration vectors of particle moving in three- dimensional space are

v v i v j v kx y z= + + ,

where vx = dx/dt, vy = dy/dt, and vz = dz/dt and

a a i a j a kx y z= + + ,

where ax = dvx /dt, ay = dvy /dt and az = dvz /dt.

P

r(t)

xj i

k

z

y

Figure 3.5


Chapter 4Laws of Motion

TARGET CBSE

1. The cause of motion is a force, which is, loosely speaking, a push or pull on the object.

2. Forces such as those that launch the basketball or pull the skier are called contact forces, because they arise from the physical contact between two objects. There are cir-cumstances, however, in which two objects exert forces on one another even though they are not touching. Such force is referred to as non-contact force or action-at- a-distance force.

3. Mass (m) can be defined as the heaviness of an object (or a body or a particle or a molecule) without gravity. Mass can also be defined as the quantity representing the amount of matter present in an object (or a body or a particle or a molecule). Mass is a scalar quantity. Mass is also known as the “inertia of a body at rest.”

4. The SI unit of force is newton (N).

5. Weight (W) is the force applied on an object due to gravi-tational attraction.

6. The magnitude of the weight (W) of an object is directly proportional to its mass (m): W m∝ .

7. Gravitational field strength (g) is the force of gravity on a unit of mass, which is a vector quantity. Weight, mass, and gravitational field strength are related as W = mg.

8. The normal reaction is a force that acts perpendicularly to a surface as a result of a force applied by an object to the surface.

9. The vector sum of the different forces acting on an object is called its net force.

10. Newton’s first law of motion: Every object continues in its state of rest or uniform motion unless made to change by a non-zero net external force.

11. The inertia of an object is its tendency to resist changes to its motion. Inertia is not a force – it is a property of all objects. The inertia of an object depends only on its mass.

12. Inertia is of three types: (a) Inertia of rest: It is the resistance of the body to change its state of rest. When a train suddenly starts, the passenger sitting tends to fall backward is an example of inertia of rest. (b) Inertia of motion: It is the resistance of the body to change its state of motion. When a bus suddenly stops, a passen-ger sitting tend to fall forward is an example of inertia of motion. (c) Inertia of direction: It is the resistance of the body to change its direction of motion. The mud from the wheels of a moving vehicle flies off tangentially is an example of inertia of direction.

13. Inertia of an object is measured by its mass. The SI unit of inertia and mass is kilogram (kg).

14. The center of mass of an object is that point through which its entire mass seems to act or seems to be concen-trated at.

15. Air resistance is the force applied to an object, opposite to its direction of motion, by the air through which it is moving.

16. Components are parts. Any vector can be resolved into a number of components. When all of the components areadded together, the result is the original vector.

17. Momentum is the product of the mass of an object and its velocity, which is a vector quantity. Momentum (

p ) of an object of mass (m) with a velocity (

v ) is expressed as

p mv= .

Momentum is also the “inertia of a body in motion.”

18. The SI unit of momentum is kg m/s.

19. Newton’s second law of motion: The rate of change of momentum of an object is directly proportional to the applied force and takes place in the direction in which the force acts. Thus, if under the action of a force F for time interval Δt, the velocity of a body of mass m changes from v to v + Δv, that is, its initial momentum p = mv changes by Δp = mΔv, according to the Newton’s second law of motion, we have

Fpt

F kpt

∝ =ΔΔ

ΔΔ

or ,

where k is the proportionality constant.

20. Impulse of a force is the product of the force and the time interval over which it acts. Impulse is a vector quantity.

21. The impulse (I) delivered by a changing force is expressed as

I = FavgΔt.

22. Newton’s third law of motion: Whenever an object applies a force (an action) on a second object, the second object applies an equal and opposite force (a reaction) on the first object Nowton’s third law is also defined as to every action there is an equal and opposite reaction and it takes place on two different bodies.

23. Law of conservation of momentum: If there are no external forces acting on a system, the total momentum remains constant, that is, if Fnet = 0, Δp = 0.

24. An object is said to be in equilibrium when it has zero acceleration.


Chapter 4 Laws of Motion14

25. A gravitational force F

g on a body is a certain type of pull that is directed toward a second body. The weight (W) of a body is equal to the magnitude Fg of the gravitational force on the body.

26. When a body presses against a surface, the surface (even a seemingly rigid one) deforms and pushes on the body with a normal force F

N that is perpendicular to the

surface.

27. Friction is the force applied on the surface of an object when it is pushed or pulled against the surface of another object.

28. There are two types of friction: (a) Static friction: The fric-tional force acting between any two surfaces at rest with respect to each other is called the force of static friction. When there is no applied force, there is no static fric-tion. It comes into play the moment there is an applied force. When the applied force increases, the static friction increases having a magnitude equal but opposite to the applied force up to a certain limit, keeping the body at rest. It is represented by fs. (b) Kinetic friction: The fric-tional force acting between surfaces in relative motion with respect to each other is called the force of kinetic friction or sliding friction. It is represented by fk.

29. Properties of friction: Property 1: If the body does not move, then the static fric-

tional force f

s and the component of F

that is parallel to the surface balance each other. They are equal in magnitude, and f

sis directed opposite that component of F

.

Property 2: The magnitude of f

s has a maximum value fs,max that is given by

fs,max = msFN,

where ms is the coefficient of static friction and FN is the magnitude of the normal force on the body from the surface. If the magnitude of the component of

F that is parallel to the surface exceeds fs,max, the body begins to slide along the surface.

Property 3: If the body begins to slide along the surface, the magnitude of the frictional force rapidly decreases to a value fk given by

fk = mkFN,

where mk is the coefficient of kinetic friction.

30. The maximum static friction that a body can exert on the other body in contact with it is called limiting friction (Fmax):

fs < Fmax = msR.

where ms is the coefficient of static friction.

31. The coefficient of static friction is always greater than the coefficient of kinetic friction, that is, ms > mk.

32. The angle of friction (f) is defined as the angle between the normal reaction N and the resultant of the friction force f and the normal reaction:

tan .φ = fN

Since f = mN, where tan f = m.

33. Angle of Repose: When a body is kept in contact with an inclined plane surface, it just starts sliding down the plane for a certain angle of inclination of the surface with the horizontal. In Fig. 4.1, the angle (a) is called the angle of repose of the inclined surface with respect to the body in contact with it a = tan-1m as

tan .a m= FR

=

amg

mg cos amg sin a

RF

a

FIGuRe 4.1

34. uniform circular motion is the motion of an object traveling at a constant (uniform) speed on a circular path. Sometimes, it is more convenient to describe uniform circular motion by specifying the period of the motion rather than the speed. The period (T) is the time required to travel once around the circle – that is, to make one complete revolution.

35. The relationship between period T and speed v is given by

vr

T= 2π

,

where 2pr is the circumference of the circle.

36. When the object is suddenly released from its circular path, the object accelerates toward the center of the circle at every moment. Such acceleration is called centri petal acceleration. The word “centripetal” means “moving toward a center.”

37. Magnitude of centripetal acceleration: The centripetal acceleration of an object moving with a speed v on a circular path of radius r has a magnitude ac given by

avrc .=

2

38. Direction of centripetal acceleration: The centripetal accel-eration vector always points toward the center of the circle and continually changes direction as the object moves.

39. An object that is in uniform circular motion can never be in equilibrium. Also an object in uniform circular motion is in dynamic equilibrium as long as the centripetal force is being counterbalanced by the centrifugal force.

40. The net force causing the centripetal acceleration is called the centripetal force ( )cF

and points in the same direc-



tion as the acceleration – that is, toward the center of the circle.

41. Magnitude of a centripetal force: The centripetal force is the name given to the net force required to keep an object of mass m, moving at a speed v, on a circular path of radius r, and it has a magnitude of

Fmv

rc .=2

42. Direction of a centripetal force: The centripetal force always points toward the center of the circle and continu-ally changes direction as the object moves.

TARGET COMPETITION

1. A part of physics is devoted to the study of motion and what can cause an object to move or accelerate. The cause of motion is a force, which is, loosely speaking, a push or pull on the object. The force is said to act on the object to change its velocity. For example, when a car accelerates, a force from the road acts on the tyre’s to cause the car’s acceleration. When a car slams into a telephone pole, a force on the car from the pole causes the car to stop.

2. The relation between a force and the acceleration it cause was first understood by Isaac Newton (1642–1727) and the study of that relation, as Newton presented it, is called Newtonian mechanics. However, Newtonian mechanics does not apply to all situations. For example, if the speeds of the interacting bodies are very large – an appreciable fraction of the speed of light – we must replace Newto-nian mechanics with Einstein’s special theory of relativity, which holds at any speed, including those near the speed of light. If the interacting bodies are on the scale of atomic structure (for example, they might be electrons in an atom), we must replace Newtonian mechanics with quan-tum mechanics.

3. Newton’s first law of motion: If no force acts on a body, the body’s velocity cannot change; that is, the body cannot accelerate. In other words, if the body is at rest, it stays at rest. If it is moving, it continues to move with the same velocity (same magnitude and same direction).

4. Force: When two or more forces act on a body, we can find their net force or resultant force, by adding the indi-vidual forces vectorially. A single force that has the mag-nitude and direction of the net force has the same effect on the body as all the individual forces together. This fact is called the principle of superposition for forces. The more proper statement of Newton’s first law is in terms of a net force:

5. Newton’s first law of motion in terms of net force: If no net force acts on a body ( ),

Fnet = 0 the body’s velocity cannot change; that is, the body cannot accelerate.

6. Inertial Reference Frames: An inertial reference frame is one in which Newton’s laws of motion (Newtonian mechanics) hold. For example, we can assume that the ground is an inertial frame provided we could neglect Earth’s astronomical motions (such as its rotation). Suppose a puck is sent sliding along a long ice strip extending from the north pole. If we view the puck from a point on the

ground so that we rotate with Earth, the puck’s path is not a simple straight line. Because the eastward speed of the ground beneath the puck is greater the farther south the puck slides, from our ground-based view the puck appears to be deflected westward. However, this appar-ent deflection is caused not by a force as required by Newton’s laws of motion but by the fact that we see the puck from a rotating frame. In this situation, the ground is a non-inertial frame.

7. Mass: Mass of a body is the characteristic that relates a force on the body to the resulting acceleration.

8. Newton’s second law of motion: The net force on a body is equal to the product of the body’s mass and its accelera-tion, where acceleration is being observed from an inertial reference frame. It can be mathematically expressed as

F manet = .

9. The acceleration component along a given axis is caused only by the sum of the force components along that same axis, and not by force components along any other axis.

10. Free-body diagram: To solve problems with Newton’s second law of motion, we often draw a free-body diagram in which the only body shown is the one for which we are summing forces. A coordinate system is usually included, and the acceleration and force on the body is shown with a vector arrow. A system consists of one or more bodies, and any force on the bodies inside the system from bodies out-side the system is called an external force.

11. Newton’s third law of motion: When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction.

12. We can call the forces between two interacting bodies a third-law force pair. These third-law pair forces are always of the same nature. When any two bodies interact in any situation, a third-law force pair is present. The third law would still hold if the interacting bodies were moving and even if they were accelerating.

13. Gravitational Force: A gravitational force

Fg on a body is a certain type of pull that is directed toward a second body. When we speak of the gravitational force

Fg on a body, we usually mean a force that pulls on it directly toward the center of Earth – that is, directly down toward the ground. Suppose a body of mass m is in free fall with the free-fall


Chapter 4 Laws of Motion16

acceleration of magnitude g. Then, if we neglect the effects of the air, the only force acting on the body is the gravitational force

Fg. We place a vertical y-axis along the body’s path, with the positive direction upward. Using Newton’s second law of motion,

F mgg .=

14. Weight: The weight of an object arises because of gravita-tional pull of Earth. Using W for the magnitude of weight, m for the mass of the object, and ME for the mass of the Earth, it follows from Newton’s law of gravitation

W GM m

r= E .2

Mass and weight are not the same quantity. Mass measures inertia whereas weight is the gravitational force acting on the object and can vary, depending on how far the object is above the Earth’s surface. The weight of an object whose mass is m depends on the values for the universal gravitation constant G, the mass ME of the Earth, and the distance r. These three parameters together deter-mine the acceleration g due to gravity. The specific value of g = 9.80 m/s2 applies only when r equals the radius RE of the Earth. When we write force due to the Earth’s attraction on a body as mg, the g here does not mean acceleration due to gravity. Symbol g is used to represent GM RE E/ .2

15. Measuring Weight: The weight W of a body is equal to the magnitude Fg of the gravitational force on the body.

One way to measure a body’s weight is to place the body on one of the pans of an equal-arm balance and then place reference bodies on the other pan until we strike a balance (so that the gravitational forces on the two sides match). The masses on the pans then match, and we know the mass of the body.

We can also weigh a body with a spring scale in which the body stretches a spring, moving a pointer along a scale that has been calibrated and marked in either mass or weight units.

The weight of a body must be measured when the body is not accelerating vertically relative to the ground. If you measure your weight with the scale in an accelerating elevator, the reading differs from your weight measured on the ground because of the acceleration. Such a meas-urement is called an apparent weight. Weight is the mag-nitude of a force and is related to mass. If you move a body to a point where the value of g is different, the body’s mass (an intrinsic property) is not different but the weight is.

16. Normal Contact Force: When a body presses against a surface, the surface (even a seemingly rigid one) deforms and pushes on the body with a normal force perpendicular to the surface.

17. Tension Force: Fig. 4.2a shows a force F being applied to the right end of a rope attached to a crate. When the cord (or a rope, cable, or other such object) is pulled taut, the cord pulls on the body with a force P directed away from the body and along the cord (Fig. 4.2a). The force is often called a tension force. We can think of a string as composed of short sections interacting by contact forces.

Each section pulls the sections to either side of it, and by Newton’s third law of motion, it is pulled by the adja-cent section. The magnitude of the force acting between adjacent sections is called tension. There is no direction associated with tension. The tension in the cord is the magnitude T of the force on the body.

(a)

(b)

(c)

F

T

−T T

FIGuRe 4.2

18. Ideal Pulley: A pulley can change the direction of the force exerted by a cord. An ideal pulley has no mass and no friction. Neither it exerts any tangential force on the cord. As a result, the tension of an ideal cord that runs through an ideal pulley is the same on both sides of the pulley. An ideal pulley can change the direction of the force exerted by a cord without changing the mag-nitude. As long as a real pulley has a small mass and negligible amount of friction, we can consider it as an ideal pulley.

Let us make a free-body diagram for a short segment of the cord at the top of the pulley (Fig. 4.3). Choosing the x-axis to be horizontal, the normal force has no x-component. Applying Newton’s second law of motion along the x-axis: ΣFx= T1cosq – T2cosq = max. For m = 0, T1 = T2. The same reasoning can be applied to any seg-ment of cord in contact with the pulley to show that the tensions are the same on either side of the pulley.

FN

T1T2

q q

Cord

Pulley

x

FIGuRe 4.3

19. Spring Force: Springs follow Hooke’s law for small extension and compression. That is, the extension or compression (the increase or decrease in length from the relaxed length) is proportional to the force applied to the



ends of the spring. Hooke’s law for an ideal spring (one which obeys Hooke’s law and is massless):

F k L= Δ .

In the above equation, F is the magnitude of the force exerted on each end of the spring and ΔL is the modulus of change in length of the spring from its relaxed length. The constant k is called the spring constant for a particu-lar spring. The spring constant is a measure of how tough it is to stretch or compress a spring. The SI unit of spring constant is N/m.

Suppose we keep an ideal spring aligned with the x-axis. One end is fixed in place and the other end can move along the x-axis (Fig. 4.4). We choose the origin so that the moveable end is at x = 0 when the spring is relaxed. Then the force is exerted by the moveable end of the spring on whatever is attached as given in Fig. 4.4. Fx is the force exerted by the moveable end when its position is x; the spring is relaxed at x = 0.

F kx= - .

The negative sign indicates the direction of the force. The moveable end of the spring always pushes or pulls toward its relaxed position.

Stretched spring

Relaxed spring

x

x = 0

FIGuRe 4.4

20. Parallel combination of springs: When springs are connected in parallel, we can replace them by a single spring of spring constant keq where keq = k1 + k2. (Fig. 4.5a).

If the force F pulls the mass m by y, the stretch in each spring will be same as y.

y1 = y2 + y

Now for an equivalent spring F = keq y and for equilibrium,

F F F k y k y k y= + ⇒ = +1 2 1 1 2 2eq

This reduces to

k k keq = +1 2

since as y1 = y2 + y. For more than two springs, we have k k k k= + + +1 2 3

Series combination of springs: When springs are connected in series, we can replace them by a single

spring of spring constant keq, where 1/keq = 1/k1 + 1/k2 (Fig. 4.5b).

F

y

k2

k1 y1F1

F2y2

m

(a)

F

(b)y

k2 k1

y1

F1F2

y2

m

FIGuRe 4.5

As the springs are massless, force in the springs will be the same

F F F1 2= = .

Now for equivalent spring F = keqy, as spring constants are not equal so extensions will not be equal, but total extension y can be written as the sum of two extensions y = y1 + y2 or

Fk

Fk

Fkeq

= +1

1

2

2

For more than two springs, we have

1 1 1

1 2k k keq

= + +

21. Motion in accelerated frames (fictitious force): Consider a block kept on the smooth surface of a compartment of train. When the train is moving at constant velocity it is an inertial frame. On an accelerating train, the block accel-erates toward the back of the train. We might conclude based on Newton’s second law of motion (F = ma) that a force is acting on the block to cause it to accelerate, but Newton’s second law of motion is not applicable from this frame. So we cannot relate observed acceleration to the force acting on the block. To use Newton’s second law of motion, we apply a fictitious force, acting in back-ward direction, that is, opposite to the acceleration of the non-inertial reference frame. The fictitious force is equal to – ma, where a is the acceleration of the non-inertial reference frame. The fictitious force appears to act on an object in the same way as a real force. However, real forces are always interactions between two objects whereas there is no second object for a fictitious force.


Chapter 5Work, Energy, and Power

TARGET CBSE

1. Although energy is often defined as the “capacity to do work,” the meaning of the term “energy” is so broad that a clear definition is difficult to write. Technically, energy is a scalar quantity associated with the state (or condition) of one or more objects.

2. Kinetic energy K is energy associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy. When the object is stationary, its kinetic energy is zero.

3. For an object of mass m whose speed v is well below the speed of light, the kinetic energy is expressed as

K mv= 12

2.

4. The SI unit of kinetic energy is joule (J).

5. Work (W) is the energy transferred to (or from) an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work.

6. The scalar product of two vectors

a and

b is written as

a b⋅ , which is defined as

a b ab⋅ = cos ,φ

where a is the magnitude of

a, b is the magnitude of

b , and f is the angle between

a and

b.

7. A dot product can be regarded as the product of two quantities: (a) the magnitude of one of the vectors and (b) the scalar component of the second vector along the direction of the first vector.

8. If the angle q between two vectors is 0°, the component of one vector along the other is maximum, and so also is the dot product of the vectors. If, instead, q is 90°, the component of one vector along the other is zero, and so is the dot product.

9. The relationship that relates work to the change in kinetic energy is known as work–energy theorem – when a net external force does work W on an object, the kinetic energy of the object changes from its initial value (KE0) to a final value (KEf), the difference between the two values being equal to the work:

W mv mv= KE KEf 0 f− = −12

12

202 .

10. Work can be expressed as follows: W ≡ FΔs cosq.

11. The SI units of work are units of force (N) times units of displacement (m) and are called joules (J):

1 J ≡ 1 N × 1 m ⇒1 J = 1 N m.

12. Work done by a constant force is expressed as W F d=

⋅ .

13. Work done by a gravitational force is expressed as Wg = mgd cosq.

14. The law of force for a spring is called Hooke’s law, which is expressed mathematically as

F kxs ,= −

where k is called the spring constant whose SI unit is the newton per meter (N/m).

15. Work done by a spring force is expressed as

W kx kxs i f .= −12

12

2 2

16. The work done by a variable force is expressed as

W F x dxx

x= ∫ ( ) .

i

f

17. Gravitational potential energy is the energy stored in an object as a result of its position relative to another object to which it is attracted by the force of gravity. Gravitational potential energy, as a function of height h, is mathematically expressed as

V(h) = mgh.

18. Elastic potential energy is the energy stored in an object as a result of a reversible change in shape, which is mathematically expressed as

V x kx( ) .= 12

2

19. Conservative force: A force is conservative when (a) the work it does on a moving object is independent of the path between the object’s initial and final positions or (b) it does no net work on an object moving around a closed path, starting and finishing at the same point.

20. Not all forces are conservative – the properties such as friction, dissipative forces, and air resistance are some examples of non-conservative force.

21. A quantity that remains constant throughout a motion is said to be “conserved.” The fact that the total mechanical energy is conserved – when the work done by the external nonconservative force (Wnc) is zero – is called the principle of conservation of mechanical energy.

22. The total of the potential energy and kinetic energy in a system is called the mechanical energy of the system.

MTPL0139 Summary Ch-5 .indd 19 4/10/2012 7:55:30 PM

Chapter 5 Work, Energy, and Power20

23. The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.

24. An internal energy that is distributed over a system on a microscopic scale is chemical energy. Reactions are called endergonic or exergonic depending on whether the system of atoms involved in the reaction ends up with more or less energy after the reaction than before.

25. The energy associated with an electric current is known as electrical energy.

26. Mass–energy equivalence: An object’s mass m and the equivalent energy E are related by the equation

E = mc2,

which is the famous Einstein’s equation.

27. Nuclear fission is the process of splitting a large nucleus to form two smaller, more stable nuclei. Fission fragments are the products that result from a nucleus that undergoes fission. The fission fragments are smaller than the original nucleus.

28. Nuclear fusion is the process of joining two smaller nuclei together to form a larger, more stable nucleus. Fusion occurs only under very extreme conditions with incredibly high temperatures and pressures, such as those inside the Sun.

29. Law of conservation of energy: Energy can neither be created, nor destroyed, that is, energy may be transformed from one form to another form, but the total energy of an isolated system remains constant.

30. The time rate at which work is done by a force is said to be the power due to the force.

31. If a force does an amount of work W in an amount of time Δt, the average power due to the force during that time interval is

PW

tavg =Δ

.

32. The instantaneous power P is the instantaneous time rate of doing work, which can be expressed as

PdWdtinst = .

33. The SI unit of power is J/s. This unit is used so often that it has a special name, watt (W), named after the scientist James Watt. 1 horse power, another unit of power often used in automobile industry, is equal to 746 W.

34. Instantaneous power is also expressed in terms of force and velocity as

P F v= ⋅

.

35. In a collision, when the kinetic energy of the system is conserved (i.e., kinetic energy is the same before and after the collision), then such a collision is called an elastic collision.

36. In a collision in which the kinetic energy of the system is not conserved, such a collision is called an inelastic collision.

37. The inelastic collision of two bodies always involves a loss in the kinetic energy of the system. The greatest loss occurs if the bodies stick together, in which case the collision is called a completely inelastic collision.

38. We call the magnitude that measures the capacity that a body has to produce an effect on other bodies in a collision linear momentum.

39. The phenomenon in which a particle system does not receive an external impulse, its total linear momentum remains constant is called principle of conservation of linear momentum.

40. The variation of linear momentum is called impulse – when the linear momentum of a body is increased, it is receiving a positive impulse; when the linear momentum of a body is decreased, the impulse is negative.

1. Although the word energy does not possess an obvious definition, it is generally explained as the capacity to do work. Technically, energy is a scalar quantity associated with the state (or condition) of one or more objects. The principle that energy can be transformed from one type to another and transferred from one object to another, but the total amount is always the same, is called law of conservation of energy.

2. Kinetic energy: Kinetic energy K is energy associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy. When the object is stationary, its kinetic energy is zero. For an object of mass m(whose speed v is well below the speed of light), the kinetic energy is expressed as

K mv= 12

2.

3. The SI unit of kinetic energy (and every other type of energy) is the joule (J), named for James Prescott Joule, an English scientist of the 1800 s.

4. Work: If you accelerate or decelerate an object by applying a force to the object, you increase or decrease the kinetic energy, K, of the object, respectively. Here transfer of energy takes place between you and the object and in such a transfer of energy via a force, work W is said to be done on the object by the force. We define work on a system by an agent exerting a force F as scalar product of F and very small displacement ds of point of application of force.

dW F ds= ⋅

.

5. Calculation of work for a uniform force: Total work done is

W F ds= ⋅∫

.

TARGET COMPETITION



For uniform force, F can be taken out of integral. Hence, W F ds= ∫

. Thus W F s= ⋅

(this is true only for uniform forces)

W F s= ( cos ),θ

where scosθ is component of displacement along force. Alternately, we can write

W F s Fs= ( cos ) cos ,θ θ=

where F cosθ is component of force along displacement.

6. The relation between work and kinetic energy for uniform forces is given by the expression

12

12

202mv mv F dx− = ,

where Fx d, which is the work, W, done on the bead by the force.

7. Work has the SI unit joule, the same as kinetic energy. An equivalent unit is newton meter (N m).

8. Work – kinetic energy theorem: Let ΔK be the change in the kinetic energy of the object, and let W be the net work done on it. Then

ΔK K K W= − =f i ,

which says that

Change in the kineticenergy of a particle

net work done onthe

=pparticle

It can also be written as

kinetic energy afterthe net work is done

kinetic energybefo

=rre the net work

the network done

+

These statements are known as the work – kinetic energy theorem for particles. They hold for both positive and negative work: If the net work done on a particle is positive, the particle’s kinetic energy increases by the amount of the work. If the net work done is negative, the particle’s kinetic energy decreases by the amount of the work.

9. Work done by the gravitational force is expressed as

W mgd mgd mgdg cos ( ) .= ° = + = +0 1

The plus sign tells us that the gravitational force now transfers energy in the amount mgd to the kinetic energy of the object. This is consistent with the speeding up of the object as it falls.

10. Work done in lifting and lowering an object: Suppose we lift a particlelike object by applying a vertical force, F to it. During the upward displacement, our applied force does positive work Wa on the object while the gravitational force does negative work Wg on it. Our applied force tends to transfer energy to the object while the gravitational force tends to transfer energy from it. The change ΔK in the kinetic energy of the object due to these two energy transfers is

ΔK K K W W= − = +f i a g.

This equation also applies if we lower the object, but then the gravitational force tends to transfer energy to the object while our force tends to transfer energy from it. When you lift a book from the floor to a shelf, both Kf and Ki are zero and the above equation reduces to

W W W Wa g a g.+ = ⇒ =0 −

Thus the work done by the applied force is the negative of the work done by the gravitational force; that is, the applied force transfers the same amount of energy to the object as the gravitational force transfers from the object.

11. Spring force: Figure 5.1 shows a spring, with one end fixed and a block attached to the other, in three different states – (a) relaxed, (b) stretched, and (c) compressed states. When the spring is stretched or compressed, it applies force to restore the relaxed state and hence spring force is sometimes called as restoring force.

Blockattachedto spring

x0

x = 0Fx = 0

(a)

x

0x

x positiveFx negative

(b)

dFs

x

0x

x negativeFx positive

(c)

dFs

FiGurE 5.1

12. To a good approximation for many springs, the force Fx

from a spring is proportional to the displacement d of the free end from its position when the spring in the relaxed state. The spring force is given by

F kds ,= −

which is known as Hooke’s law after Robert Hooke, an English scientist of the late 1600s.The minus sign indicates that the direction of the spring force is always opposite the direction of the displacement of the spring’s free end. The constant k is called the spring constant (or force constant) and is a measure of the stiffness of the spring. Considering xaxis, we have

F kxx = − .


Chapter 5 Work, Energy, and Power22

Note that a spring force is a variable force because it is a function of x, the position of the free end.

13. The SI unit for spring constant k is newton per meter (N/m).

14. Work done by a spring force is expressed as

W kx kxs f ,= −12

12

2 2i

where the work, Ws, is positive if the block ends up closer to the relaxed position (x = 0) than it was initially. It is negative if the block ends up farther away from x = 0. It is zero if the block ends up at the same distance from x = 0.

15. Work done by an applied force: When we displace the block along the x-axis while continuing to apply a force Fa to it. During the displacement, our applied force does work Wa on the block while the spring force does work Ws. The change ΔK in the kinetic energy of the block due to these two energy transfers is

ΔK K K W W= − = +f i a s .

If a block that is attached to a spring is stationary before and after a displacement, that is, both Kf and Ki are both zero, the work done on it by the applied force displacing it is the negative of the work done on it by the spring force.

16. Work for a non-uniform force (a variable force) in one dimension is expressed as

W F x dxx

x= ∫ ( ) .

t

f

17. Work for a non-uniform force (a variable force) in three dimensions is expressed as

W dWF F dx F dy F dzr

r

xx

x

yy

y

zz

z= = + +∫ ∫ ∫ ∫

i i i i

f f f f.

18. Validity of work – kinetic energy theorem in inertial reference frames: Two observers in two different inertial frames will observe the same force but different displacement for a given particle. Thus, they will measure different values for the work done on the particle.

19. Work – kinetic energy theorem for a system (collection of particles): While verifying work – kinetic energy theorem for a system of many particles, we must remember that work done by all the forces (external and internal) must be considered. Otherwise the equation should be used separately on individual particles. Although total work done by static friction, tension, and normal contact force that is by action and reaction on a system will always add up to zero.

20. Potential energy is the energy that can be associated with the configuration (arrangement) of a system of objects that exert forces on one another.

21. A gravitational potential energy is associated with the state of separation between two objects (an object – Earth system) that attract each other by the gravitational force:

U h mgh( ) ,=

where h is the vertical position (height) of the object.

22. An elastic potential energy is associated with the state of compression or extension of an elastic object, here the bungee cord:

U x kx( ) .= 12

2

23. Work and potential energy are related as follows:

ΔU W= − ,

in which, for either rise or fall, the change ΔU in gravitational potential energy is defined as being equal to the negative of the work done on the tomato by the gravitational force.

24. Conservative forces: Let the work done by a force on an object, transferring energy between the kinetic energy K of the object and some other type of energy of the system be W1 and when the force reverses, let the work done be W2. Then, in a situation in which W1 = –W2 is always true and the other type of energy is a potential energy, the force is said to be a conservative force. The gravitational force and the spring force are both conservative forces. A force that is not conservative is called a nonconservative force. The kinetic frictional force and drag force are nonconservative.

25. Work – mechanical energy theorem: From work – kinetic energy theorem it is clear that if some forces act on a body, the sum of the work done by individual forces acting on it is equal to the change in kinetic energy. If some of them are conservative and others are nonconservative, for conservative forces, we can write potential energy as follows:

W W K K

U U W K K

W K K U U

c nc f i

f t nc f i

nc f i f i

{ ( )}

( ),

+ = −

− − + = −

= − + −

∑∑∑ ∑

∑ ∑ In which the term on RHS is often called mechanical

energy; to conclude, effect of a force can either be written as work on LHS or it can come as potential energy on RHS.

26. Conservation of mechanical energy is expressed as

K U K U2 2 1 1+ − + − .

In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy Emechanical of the system cannot change. This result is called the principle of conservation of mechanical energy.

Δ Δ ΔE K Umechanical .= + = 0

When the mechanical energy of a system is conserved, we can relate the sum of kinetic energy and potential energy at one instant to that at another instant without consider-ing the intermediate motion and without finding the work done by the forces involved.

27. Work done on a system by an external force – when no friction involved – is expressed as

W K U W E= + ⇒ =Δ Δ Δ mechanical ,



where ΔEmechanical is the change is the mechanical energy of the system.

28. Work done on a system by an external force – when friction involved – is expressed as

W E E= +Δ Δmechanical thermal ,

where ΔEmechanical is the change is the mechanical energy of the system and ΔEthermal is the change is the thermal energy of the system.

29. Conservation of energy: We know that energy obeys the law of conservation of energy, which is concerned with the total energy E of a system. That total is the sum of the system’s mechanical energy, thermal energy, and any type of internal energy in addition to thermal energy. The total energy E of a system can change only by amounts of energy that are transferred to or from the system. The only type of energy transfer that we have considered is work W done on a system. Thus, for us at this point, this law states that

W E E E E= = + +Δ Δ Δ Δmechanical thermal internal ,

where ΔEmechanical is any change in the mechanical energy of the system, ΔEthermal is any change in the thermal energy of the system, and ΔEinternal is any change in any other type of internal energy of the system. Included in ΔEmechanical are changes ΔK in kinetic energy and changes ΔU in potential energy.

30. isolated system: If a system is isolated from its environment, there can be no energy transfers to or from it. For that case, the law of conservation of energy states that the

total energy E of an isolated system cannot change. For an isolated system,

Δ Δ ΔE E Emechanical theraml+ + =intern l .a 0

In an isolated system, we can relate the total energy at one instant to the total energy at another instant without considering the energies at intermediate times.

31. External forces and internal energy transfers: An external force can change the kinetic energy or potential energy of an object without doing work on the object – that is, without transferring energy to the object. Instead, the force is responsible for transfers of energy from one type to another inside the object.

32. Power: Power is defined as the rate at which work is done by a force. If an amount of energy ΔE is transferred in an amount of time Δt, the average power due to the force is

PEtavg = Δ

Δ.

Similarly, the instantaneous power due to the force is

PdEdt

= .

33. instantaneous power is given by

PdWdt

FdRdt

F vI . .= = = ⋅

34. Average power is given by

Average power Work done by a force

Time taken to do work = ==

ΔΔWt

.


Chapter 6System of Particles and Rotational Motion

TARGET CBSE

1. A rigid body is a system of particles in which the distances between the particles do not vary with time. If an object that is solid enough that it has a fixed shape throughout its motion, it is considered a perfectly rigid body.

2. Many objects have an axis of symmetry, a line about which the object may be turned and still look the same, like the line passing through the center of a cylinder or a ball.

3. All spinning objects display rotational motion.

4. When a rigid body is in rotation about a fixed axis, the axis is called the axis of rotation or the rotational axis.

5. When the rotational axis of a rigid body in rotation does not move, it is called fixed axis.

6. A reference line is perpendicular to the axis of rotation so that it lies in the x–y plane. The reference line is fixed with respect to the rotating body so that it rotates around the z-axis as the body rotates.

7. All particles of the body at any given instant of time have the same velocity in pure translational motion.

8. Every particle of the rigid body moves in a circle, which lies in a plane perpendicular to the axis and has its center on the axis in case of rotation of a rigid body about a fixed axis.

9. The motion of a rigid body which is not pivoted or fixed in some way is either a pure translation or a combination of translation and rotation.

10. The motion of a rigid body which is fixed or pivoted in some way is purely rotational.

11. The center of mass of a system of particles is the point that moves as though (1) all of the system’s mass were concentrated there and (2) all external forces were applied there. The center of mass of a system of particles is defined as the point whose position vector is expressed as

rM

m ri

n

i icom = ∑=

11

,

where M is the total mass of the system.

12. The cross product of a vector with itself is a null vector.

A A A A n× = ° =( )( )sin .0 0

13. The cross product of two vectors does not obey commu-tative law.

A B B A× ≠ × .

14. The cross product obeys the distributive law.

A B C A B A C× + = × + ×( ) .

15. The (instantaneous) magnitude of angular velocity w is expressed as

ω θ= ddt

.

16. Angular velocity is a vector quantity and it is directed along axis of rotation.

17. The unit of angular velocity is commonly the radian per second (rad/s) or the revolution per second (rev/s).

18. The relationship between angular velocity and linear velocity is expressed as

v = rw.

19. The linear momentum of a particle is a vector quantity that is defined as

p mv= ,

in which m is the mass of the particle and

v is its velocity.

20. If

a and

b are parallel or antiparallel,

a b× = 0. The mag-nitude of

a b× , which can be written as | |,

a b× is maxi-mum when

a and

b are perpendicular to each other.

21. The angular momentum of a system of particles is given by

L r p= × .

22. The torque

τ acting on the particle relative to the fixed point O is a vector quantity, which is defined as

τ = ×r F .

23. Conditions for equilibrium: (1) Resultant of all the exter-nal forces ( )Fnet and external torques ( )τnet must be zero. (2) Centre of gravity is the location in the extended body where we can assume the whole weight of the body to be concentrated.

24. The gravitational force

Fg on a body that effectively acts at a single point is called the center of gravity of the body.

25. The moment of inertia of a rigid body about an axis is expressed as

I m ri i= ∑ 2 ,

where ri is the perpendicular distance of the ith point of the body from the axis. The kinetic energy of rotation is given by

K I= 12

2ω .


Chapter 6 System of Particles and Rotational Motion26

26. The SI unit of moment of inertia is kilogram square meter (kg m2).

27. Theorem of parallel axes: The moment of inertia of a body about any axis is equal to the sum of the moments of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of distance between the two parallel axes:

I I Md= +g .2

28. Theorem of perpendicular axes: The moment of inertia of planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two per-pendicular axes concurrent with perpendicular axis and lying in the plane of the body:

I I Iz x y= + .

29. In terms of dynamics and kinematics, rotation about a fixed axis is analogous to linear motion.

30. The angular acceleration of a rigid body which is rotating about a fixed axis is expressed as

Ia = t.

If the external torque t is zero, the component of angu-lar momentum about the fixed axis Iw of such a rotating body is constant.

31. For rolling motion without slipping vcom = rw, where vcom is the velocity of translation, that is, of the centre of mass, r is the radius, and m is the mass of the body. The kinetic energy of such a rolling body is the sum of kinetic ener-gies of translation and rotation:

K I Mv= +12

2 12

2com comω .

32. Law of conservation of angular momentum: If the net resultant external torque acting on an isolated system is zero, the total angular momentum L of system should be conserved.

33. The relation between the arc length s covered by a par-ticle on a rotating rigid body at a distance r from the axis and the displacement q (in radians) is expressed as

s = rq.

TARGET COMPETITION

1. The center of mass of a system of particles is the point that moves as though (1) all of the system’s mass were concen-trated there and (2) all external forces were applied there. The center of mass of a system of particles is represented in vector form as

rM

m ri

n

i icom = ∑=

11

.

2. Newton’s second law for a system of particles is expressed as

F Manet com= ,

where

Fnet is the net force of all external forces that act on the system, M is the total mass of the system. We assume that no mass enters or leaves the system as it moves, so that M remains constant. The system is said to be closed.

acom

is the acceleration of the center of mass of the system.

3. Linear momentum of a particle is a vector quantity .p that is, defined as

p mv= ,

in which m is the mass of the particle and v is its velocity.

4. Newton expressed his second law of motion in terms of momentum as follows: The time rate of change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of that force:

Fdpdt

ddt

mv mdvdt

manet = = = =( ) .

5. Linear Momentum of a system of particles: Consider a system of n particles, each with its own mass, velocity, and linear momentum. The particles may interact with each other, and external forces may act on them. The system as a whole has a total linear momentum p, which is defined

to be the vector sum of the individual particles’ linear momenta. Thus,

P p p p p

m v m v m v m v

P Mv

n

n n

= + + + += + + + +

=

1 2 3

1 1 2 2 3 3

com.

The linear momentum of a system of particles is equal to the product of the total mass M of the system and the velocity of the center of mass.

6. A rigid body may be defined as a system of particles in which the distances between the particles do not vary with time. Majority of solid objects that exist in nature can be called “rigid” as they deform so little in shape and size under ordinary conditions and these deformations may be entirely neglected for applying laws of motion on the body.

7. Translation motion:

a. In a pure translation (linear motion), every point of the body moves in a straight line, and every point moves through the same linear distance during a particular time interval. Here, all the particles forming the body move along parallel paths. If these paths are straight lines, the motion is said to be a rectilinear translation and if the paths are curved lines, the motion is a curvilinear translation. In translation motion, all the points on the solid traverse equal distances in the same time interval. Therefore, velocities, as well as accelerations, of all points on the body at the given moment of time can be described by the same vectors. This allows the study of translation of a solid to be simplified to the study of motion of an individual point on that body.

b. Rotation about a fixed axis: Rotation is simply the turning motion of an object about an axis. In a pure



rotation (angular motion), every point of the body moves in a circle whose center lies on the axis of rotation, and every point moves through the same angle during a particular time interval. The fixed axis of rotation is called the axis of rotation (may be outside or inside the rigid body). If this axis, intersects the rigid body, the particles located on the axis have zero velocity and hence zero acceleration.

8. Angular position: A reference line fixed in the rigid body is considered to describe angular position. The angular posi-tion of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position. The angular position q is measured relative to the positive direction of the x-axis. q is given by

θ = sr

,

where s is the length of the circular arc and r is the radius of the circle. An angle defined in this way is measured in radians (rad) rather than in revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension:

1 3602

2rev rad.= ° = =p prr

9. Angular displacement Dq of the body from position q1 to q2 is given by

∆θ θ θ= −2 1.

The displacement ∆x is either positive or negative, depending on whether the body is moving in the positive or negative direction of the axis. Similarly, an angular dis-placement in the counterclockwise direction is positive, and one in the clockwise direction is negative.

10. Average angular velocity of the body in the time interval ∆t from t1 to t2 is defined as

ω θ θ θavg = −

−=2 1

2 1t t t∆∆

in which ∆q is the angular displacement that occurs during ∆t, angular position q1 at time t1 and at angular position q2 at time t2. The (instantaneous) angular velocity w, is the limit of the ratio as ∆t approaches zero. Thus,

ω θ θ= ∆∆

=∆ →lim .

t tddt0

11. The angular velocity w of a rotating rigid body is either positive or negative, depending on whether the body is rotating counterclockwise (positive) or clockwise (nega-tive). The magnitude of instantaneous angular velocity is equal to the angular speed.

12. The average angular acceleration of the rotating body in the interval from t1 to t2 is defined as

α ω ω ωavg = −

−= ∆

∆2 1

2 1t t t

in which ∆w is the change in the angular velocity that occurs during the time interval ∆t. The (instantaneous)

angular acceleration a, is the limit of this quantity as ∆t approaches zero. Thus,

α ω ω= ∆∆

=∆ →lim

t tddt0

13. Position: If a reference line on a rigid body rotates through an angle q, a point within the body at a position r from the rotation axis moves a distance s along a circular arc, where s is given by

s r= θ ,

14. The speed: Differentiating position equation with respect to time – with r held constant – leads to

dsdt

ddt

r=θ

,

where ds/dt is the linear speed (the magnitude of the linear velocity) of the point in question, and dq/dt is the angular speed w of the rotating body. So

v r= ω ( ).radianmeasure

Since all points within the rigid body have the same angu-lar speed w, points with greater radius r have greater lin-ear speed v. Also linear velocity is always tangent to the circular path of the point in question. Each point within the body undergoes uniform circular motion. The period of revolution T for the motion of each point and for the rigid body itself is given by

Tr

v= 2π

.

15. Kinetic energy of rotation: Treating the rotating rigid body as a collection of particles with different speeds, we obtain, the kinetic energy of a rotating body as,

K m v m v m v

M V

= + + +

= ∑

12

12

12

12

1 12

2 22

3 32

1 12

in which mi is the mass of the i th particle and vi is its speed. The sum is taken over all the particles in the body. Substi-tuting v = w r,

K m r m r= ( ) = ( )∑∑ 12

121 2

2

1 12 2ω ω

in which w is the same for all particles. m r1 12∑ tells us how

the mass of the rotating body is distributed about its axis of rotation. We call that quantity the rotational inertia (or moment of inertia)I of the body with respect to the axis of rotation. It is a constant for a particular rigid body and a particular rotation axis. Therefore,

I m r= ∑ 1 12 ( )rotational inertia

Kinetic energy associated with rotational motion is

K I= 12

2ω ( )radianmeasure

The SI unit for I is the kilogram square meter (kg m2).


Chapter 6 System of Particles and Rotational Motion28

16. Parallel-axis theorem: Suppose we want to find the rota-tional inertia I of a body of mass M about a given axis. Let Icom (known value) be the rotational inertia of the body about a parallel axis that extends through the body’s center of mass and let h be the perpendicular distance between the given axis and the axis through the center of mass (remember these two axes must be parallel). Then, the rotational inertia I about the given axis is

I I Mh= +com .2

17. Perpendicular axis theorem: The moment of inertia Ix , Iy , Iz of mass point mi along x, y, z axes are

I m x

I m y

I m r

x i i

y i i

z i i

= ∑

= ∑

= ∑

2

2

2

;

;

.

On adding the moment of inertia about two axes perpen-dicular to the given axis, moment of inertia of given axis can be found, that is,

I I m x y m r Ix y z+ = + = =∑ ∑1 12

12

1 12( ) .

18. Torque, which comes from the Latin word meaning “to twist,” may be loosely identified as the turning or twisting action of the force F

. Torque is defined by the equation

τ φ= ( )( sin ).r F

The SI unit of torque is the newton meter (N m). Torques obey the superposition principle for forces: When several

torques act on a body, the net torque (or resultant torque) is the sum of the individual torques. The symbol for net torque is τnet .

19. The angular momentum of a particle with respect to the origin is a vector quantity defined as

L r p m r v= × = ×( ) ),(angular momentum defined

where r is the position vector of the particle with respect to origin O. As the particle moves relative to origin O, the position vector r rotates around O. Angular momentum bears the same relation to linear momentum that torque does to force. The SI unit of angular momentum is kilogram-meter-squared per second (kg m2/s), equivalent to the joule-second (J s).

20. If no net external torque acts on the system, Newton’s second law in angular form

τnet /= dL dt , becomes

L

= a constant (isolated system).

This result, called the law of conservation of angular momentum, can also be written as (net angular momen-tum some initial time ti) = (net angular momentum at some final time tf):

L Li

= f ( ).isolated system

If the net external torque (or component along an axis) acting on a system is zero, the angular momentum L (or component along that axis) of the system remains constant, no matter what changes take place within the system.


Chapter 7Gravitation

TARGET CBSE

1. Newton concluded that not only Earth attracts both apples and the Moon but also every body in the universe attracts every other body; this tendency of bodies to move toward one another is called gravitation.

2. Newton proposed a force law that we call Newton’s law of gravitation is defined as every particle attracts other particle with a gravitational force of magnitude

F Gm m

r= 1 2

2 ,

where m1 and m2 are the masses of the particles, r is the distance between them, and G is the gravitational con-stant (= 6.672 ×10–11 Nm2/kg2 or = 6.67 ×10–11 m3/kg s2).

3. Gravitational constant G is numerically equal to the force of attraction between unit mass separated by a unit dis-tance. The value of G was found out by Cavendish.

4. The SI unit of gravitational constant is N/kg2/m2 and its dimensional formula is [M−1L3T−2].

5. Shell theorem: A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

6. If we are supposed to find the resultant gravitational force acting on the particle m due to a number of masses M1, M2, … Mn, we use the principle of superposition. Let F1, F2,…Fn be the individual forces due to the masses M1, M2,….Mn, which are given by the law of gravitation, then from the principle of superposition, each of these forces acts independently and uninfluenced by the other bodies. The resultant force FR can be expressed in vector addition as

F F F F Fii

n

R n= + + + ==∑1 2

1

,

where ∑ is the symbol used for summation.

7. Weight is the force of gravity acting on an object. One’s weight on the surface of the Earth differs from the surface of the Moon or any other celestial body, and in the depths of interstellar space, one may weigh next to nothing. Weight a body is related to its mass, which is the amount of matter in the body. However, weight is different from mass. In interstellar space, the weight of a body would be zero, but its mass would not change.

8. Kepler’s first law of planetary motion: All planets move in elliptical orbits with the Sun at one of the focal points, which is also called law of orbit.

9. Kepler’s second law of planetary motion: The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals, which is also called law

of area. This law follows from the fact that the force of gravitation on the planet is central and hence angular momentum is conserved.

10. Kepler’s third law of planetary motion: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the elliptical orbit of the planet, which is also called law of period, is given by

TGM

R22

34=

π

s

,

where T is the period of motion of the planet, R is the radius of the circular orbit of the planet, Ms is the mass of the Sun, and G is the universal gravitational constant (= 6.672 ×10–11 Nm2/kg2). For elliptical orbits, this equation is valid if R is replaced by the semi-major axis (a).

11. When a body of mass m lying on the surface of the Earth of mass ME and radius RE, the exact value of acceleration due to gravity at an altitude h above the surface of Earth is then given by

gGM

R= E

E

,2

12. Considering Earth be a homogeneous sphere of radius RE and mass ME and a body be taken to a depth d below the free surface of the Earth, then the acceleration due to gravity is gd is given by

g gdRd

E

= −

1 .

13. The gravitational potential energy U of two particles, of masses M and m, separated by a distance r is given by

UGMm

r= − ,

The gravitational potential energy decreases when the separation decreases. Since U = 0 for r = ∞, the potential energy is negative for any finite separation and becomes progressively more negative as the particles move closer together.

14. When a projectile is projected upward from the Earth’s, usually it will slow, stop momentarily, and return to Earth. There is, however, a certain minimum initial speed that will cause it to move upward forever, theoretically coming to rest only at infinity. This minimum initial speed is called the (Earth’s) escape speed.

15. When an isolated system consists of a particle of mass m moving with a speed v in the vicinity of a massive body of


Chapter 7 Gravitation30

mass M, then the total mechanical energy of the particle is given by

E mvGMm

r= −1

22 ,

which implies that the total mechanical energy is the sum of the kinetic and potential energies. The total energy is a constant of motion.

16. The escape velocity, ve , of a body that is projected from the Earth is given by

v gRe ,= 2 E

which has the value of 11.2 km/s.

17. A satellite in a circular orbit around the Earth in the equa-torial plane, which appears stationary to an observer on the Earth, is called geostationary satellite.

18. Polar orbit is that orbit whose angle of inclination with equatorial plane of Earth is 90°. Polar satellites are low altitude (h ≈ 500 − 800 km) satellites, which circle the globe in a North–South orbit passing over the North and South poles. Polar satellites cross the equator at the same time daily. This is because they are sun synchronous.

19. Weight of a body is the force with which the body is attracted by the Earth toward its center (W = mg). The reduction in weight is due to the fact that the Earth’s grav-ity grows weaker with increasing altitude. Weightlessness (generally, only deep in space from any star or planet) can be achieved in the following situations: (1) Weight of a body in a freely falling lift is zero. (2) Weight of a body is zero at the point where the gravitational fields of Earth and Moon cancel out. This point is called zero-gravity point. (3) At the centre of the earth (where g = 0), the weight of a body is zero.

TARGET COMPETITION

1. The gravitational force holds together the entire universe, which is expanding. This force is also responsible for some of the most mysterious structures in the universe: black holes. When a star considerably larger than our Sun burns out, the gravitational force between all its particles can cause the star to collapse in on itself and thereby to form a black hole. The gravitational force at the surface of such a collapsed star is so strong that neither particles nor light can escape from the surface (thus the term “black hole”). Any star coming too near a black hole can be ripped apart by the strong gravitational force and pulled into the hole. Enough captures like this yields a supermassive black hole.

2. Newton’s law of gravitation: In 1665, Isaac Newton made a basic contribution to physics when he showed that the force that holds the Moon in its orbit is the same force that makes an apple fall. He concluded that every body in the universe attracts every other body; this tendency of bodies to move toward each other is called gravitation. He proposed a force law that we call Newton’s law of gravitation: Every particle attracts any other particle with a gravitational force of magnitude

F Gm m

r= 1 2

2 ,

where m1 and m2 are the masses of the particles, r is the distance between them, and G is the gravitational con-stant, with a value that is now known to be

G = × −6 67 10 11. .Nm /kg2 2

3. Newton solved the apple–Earth problem by proving an important theorem called the shell theorem which states that uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were con-centrated at its center.

4. Given a group of particles, we find the net (or resultant) gravitational force on any one of them from the others by using the principle of superposition. This is a general principle that says a net effect is the sum of the individual effects. For n interacting particles, we can write the prin-ciple of superposition for the gravitational forces on par-ticle 1 as

F F F F F F n

1,net = + + + +⋅⋅⋅+12 13 14 15 1 ,

where F1,net is the net force on particle 1 and, say, F13 is the force on particle 1 from particle 3. Expressing this equa-tion as a vector sum, we get

F F i

i

n

1,net ==∑ 1

2

.

5. Gravitation near Earth’s surface: Let us assume that Earth is a uniform sphere of mass M. The magnitude of the gravitational force from Earth on a particle of mass m, located outside Earth at distance r from Earth’s center, is then given by

F GMm

r= 2 .

If the particle is released, it will fall toward the center of Earth, as a result of the gravitational force F, with an accele-ration called the gravitational acceleration, ag , which is given by

aGMrg = 2 , (1)

ag varies negligibly with altitude. Any g value measured at a given location will differ from the ag value, for that loca-tion, for three reasons, namely, (1) Earth’s mass is not distributed uniformly, (2) Earth is not a perfect sphere, and (3) Earth rotates.



6. Gravitation inside Earth: If Earth’s mass were uniformly distributed, the gravitational force acting on a parti-cle would be a maximum at Earth’s surface and would decrease as the particle moved outward, away from the planet. If the particle were to move inward the gravita-tional force would change for two reasons: (1) It would tend to increase because the particle would be mov-ing closer to the center of Earth and (2) it would tend to decrease because the thickening shell of material lying outside the particle’s radial position would not exert any net force on the particle.

For a uniform Earth, the second influence would prevail and the force on the particle would steadily decrease to zero as the particle approached the center of Earth. How-ever, for the real (non uniform) Earth, the force on the par-ticle actually increases as the particle begins to descend. The force reaches a maximum at a certain depth and then decreases as the particle descends farther.

7. Gravitational potential energy: Consider the gravita-tional potential energy U of two particles, of masses m and M, separated by a distance r. We choose a reference con-figuration with U equal to zero. The separation distance r in the reference configuration is taken large enough (to simplify the equations) to be approximated as infinite. The gravitational potential energy decreases when the separation decreases. Since V = 0 for r = ∞, the potential energy is negative for any finite separation and becomes more negative as the particles move closer together. We take the gravitational potential energy of the two-particle system to be

U rGMm

r( ) = − .

Note that U(r) approaches zero as r approaches infin-ity and that for any finite value of r, the value of U(r) is negative. If the system contains more than two particles (say m1, m2, m3), we consider each pair of particles in turn, the potential energy of the system as

UGm m

rGm m

rGm m

r= − + +

1 2

12

1 3

13

2 3

23

.

8. Path independence: The change ∆U in the gravitational potential energy of a body from the initial point i to the final point f is given by

∆U V V W= − = −f i .

Since the work W done by a conservative force is inde-pendent of the actual path taken, the change ∆U in gravi-tational potential energy is also independent of the path taken.

9. Potential energy and force: We can first find the expres-sion for potential energy function and then derive the force function. From F(x) = – dU(x)/dx, we can write

FdUdr

ddr

GMmr

= − = − −

= − GMmr 2 ,

which is Newton’s law of gravitation. The minus sign indi-cates that the force on mass m points radially inward, toward mass M.

10. Escape speed: An object thrown or fired upwards at a certain minimum initial speed will move upward forever, theoretically coming to rest only at infinity. This minimum initial speed is called the (Earth’s) escape speed.

Consider a projectile of mass m, leaving the surface of a planet of mass M and radius R(or some other astronomical body or system) with escape speed v. The projectile has a kinetic energy K given by ½mv2 and a potential energy V given by U = –GMm/R. When the projectile reaches infin-ity, it stops and thus has no kinetic energy. It also has no potential energy because an infinite separation between two bodies is our zero-potential-energy configuration. Its total energy at infinity is therefore zero. From the principle of conservation of energy, its total energy at the planet’s surface (initial total energy) must also have been zero, and so

K U mvGMm

R+ = + −

=12

02 ,

that is

vGMR

= 2.

The escape speed of a projectile from any astronomical body can be found, provided we substitute the mass of the body for M and the radius of the body for R. Attaining the escape speed is easier if the projectile is fired in the direction the launch site is moving as the planet rotates.

11. Planets and satellites – Kepler’s laws Kepler’s first law or law of orbits: All planets move in

elliptical orbits, with the Sun at one focus. When a planet of mass m moving in such an orbit around the Sun, whose mass is M and we assume that M >> m, so that the center of mass of the planet–Sun system is approximately at the center of the Sun, the orbit in is described by giving its semi-major axis a and its eccentricity e such that ea is the distance from the center of the ellipse to either focus F or F′. An eccentricity of zero corresponds to a circle, in which the two foci merge to a single central point. The eccentricities of the planetary orbits are not large. For Earth’s orbit, e = 0.0167.

Kepler’s second law or law of areas: A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals; that is, the rate dA/dt at which it sweeps out area A is constant. The second law tells us that the planet will move most slowly when it is farthest from the Sun and most rapidly when it is nearest to the Sun. Kepler’s second law is equivalent to the law of conservation of angular momentum.


Chapter 7 Gravitation32

Kepler’s second law or law of periods: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit. Considering a planet of mass m moving around the Sun of mass M in a circular orbit of radius r (the radius of a circle is equivalent to the semi-major axis of an ellipse). Applying Newton’s second law of motion (F = ma) to the orbiting planet yields

GMmr

m r22= ( )( )ω

Here we have substituted w2r for the centripetal accelera-tion. Now w = 2p / T, where T is the period of the motion. Substituting this value we obtain Kepler’s third law:

TGM

r22

34=

π.

The quantity in parentheses is a constant that depends only on the mass M of the central body about which the planet orbits. Equation 34 holds also for elliptical orbits, provided we replace r with a, the semi-major axis of the

ellipse. This law predicts that the ratio T 2/a3 has essentially the same value for every planetary orbit around a given massive body.

12. Satellites: Orbits and energy: As a satellite orbits Earth in an elliptical path, the kinetic energy K and gravitational potential energy U, fluctuate with fixed periods owing to change in its speed and distance respectively. However, the mechanical energy E of the satellite remains constant (since the satellite’s mass is so much smaller than Earth’s mass, we assign U and E for the Earth–satellite system to the satellite alone), which is given by

EGMm

r= −

2 (circular orbit)

and

EGMm

a= −

2 (elliptical orbit)

The total energy of an orbiting satellite depends only on the semi-major axis of its orbit and not on its eccentricity e.


Chapter 8Mechanical Properties of Solids

TARGET CBSE

1. Solids have a definite shape and size – a force is required to deform their shape and size. For example, the length of a rubber band increases when we stretch the rubber band by applying some force on it. When the force applied on the rubber band is removed, the rubber band regains its original configuration.

2. Elasticity is the property of a body by virtue of which it tends to regain its original size and shape after the applied force is removed. The deformation caused in the body is known as elastic deformation. Examples of elastic mate-rials are quartz fiber, phosphor bronze, etc.

3. Plasticity is the inability of a body to regain its original status on the removal of the deforming forces. Examples of plastic materials are Bakelite, plastic, etc.

4. The internal restoring force acting per unit area of a deformed body is called stress.

5. When we apply a deforming force on an elastic body, it produces a change in the configuration of the body. The change in configuration per unit original configuration is known as strain.

6. The SI unit of stress is same as that of pressure, that is, N/m2 or pascal (Pa) and its dimensional formula is [ ].ML T1 2− −

7. If F is the magnitude of the force applied on the body and A is the area of cross-section of the body, the magnitude of stress is given by

StressRestoring force

Area= = F

A.

8. When the elastic restoring force or deforming force acts perpendicular to the cross-sectional area of the body on which deforming force is applied, the stress is called normal stress.

a. When there is an increase in the length or the extension of the body in the direction of the force applied, the stress set up is called tensile stress.

b. When there is a decrease in the length or the compres-sion of the body due to the deforming force applied nor-mally on the body, the stress set up is called compressive stress.

9. Tensile stress and compressive stress can also be termed as longitudinal stress because both produce a change in the length of the body being deformed.

10. The restoring force per unit area developed in a body due to the applied tangential force is known as tangential stress or shearing stress. Mathematically, it is expressed as

Shearing stressTangential force

Area= .

11. Pressure is the force per unit area. The strain is ΔV/V, where V is the original volume of the specimen and ΔV the absolute value of the change in volume. The correspond-ing modulus, with symbol B, is called the bulk modulus of the material. The object is said to be under hydraulic compression and the pressure can be called the hydraulic stress.

12. Solids, in general – with their rigid atomic lattices – are less compressible than liquids, in which the atoms or mole cules are less tightly coupled to their neighbors.

13. The ratio of change in configuration to original configura-tion is called strain. Mathematically, strain is written as

Strain = Change in configuration

Original configuration.

14. If the deforming force acting on an elastic body produces a change in only the length of the body, the change in length per unit original length of the body is known as longitudi-nal strain, which is mathematically expressed as

Longitudinal strain=Change in length

Original length= ΔL

L.

15. If the deforming force acting on an elastic body produces a change in the shape of the body without changing its volume, the strain produced in the body is known as shearing strain, which is mathematically expressed as

Shearing strain ( ) .θ = ΔLL

16. If the deforming force acting on an elastic body produces a change in the volume of the body alone, the change in volume per unit original volume of the body is known as volumetric strain, which is mathematically expressed as

Volumetric strain=Change in volume

Original volume= ΔV

V.

17. According to Hooke’s law, within elastic limits, stress is directly proportional to strain, that is, the extension pro-duced in a wire is directly proportional to the load applied to the wire, which can be expressed as

Stress StrainStress Strain,

∝= ×k

where k is a constant of proportionality and is known as the modulus of elasticity. Hooke’s law is applicable to most of the materials but there are certain materials in which the relationship between stress and strain is not linear and they do not obey Hooke’s law.


Chapter 8 Mechanical Properties of Solids34

18. The maximum load to which the wire is subjected divided by its original cross-sectional area is called the ultimate strength or the tensile strength of the wire and is some-times also termed as breaking stress.

For a wire, when the load is decreased – that is, the stress is reduced – the wire finally breaks at a point which repre-sents the breaking point.

19. Substances such as tissue of aorta and rubber which can be stretched to cause large strains are called elastomers. For elastomers, the stress–strain variation is not a straight line. The elastic region for such materials is very large and they do not obey Hooke’s law.

20. Ductile materials show large plastic range beyond elastic limit. Examples of ductile materials are copper, silver, iron, aluminum, etc.

21. Brittle materials show very small plastic range beyond elastic limit. The breaking point lies close to the elastic limit. Examples of brittle materials are cast iron, glass, etc.

22. The property of an elastic body due to which its behavior becomes less elastic under the action of repeated alternating deforming forces is called elastic fatigue. For example, on stretching and releasing the rubber again and again, the elasticity of rubber is lost.

23. Elastic after-effect is the temporary delay in regaining the original configuration by an elastic object when all the deforming forces acting on the object are removed.

24. The ratio of the stress to the corresponding strain pro-duced in a body within the elastic limits is called modulus of elasticity or coefficient of elasticity. Modulus of elas-ticity is numerically equal to the ratio of stress and strain and, therefore, it has same dimensions as stress:

Modulus of elasticity = StressStrain

.

25. Depending upon the different types of stress and strain, there are three different types of modulus of elasticity: (1) Young’s modulus of elasticity, (2) shear modulus of elasti city, and (3) bulk modulus of elasticity. We will discuss each type of modulus of elasticity in detail in this section.

26. The ratio of normal stress to the longitudinal strain within the elastic limit is called Young’s modulus of elasticity, which is mathematically expressed as

Y = =Tensile (or compressive) stressLinear strain

σε

.

Greater the Young’s modulus of a material, larger is the elasticity of the material. Therefore, steel is more elastic than copper because Young’s modulus of steel is greater than that of copper.

27. Young’s modulus of the material of a wire is expressed as

YMg r

L LMgLr L

= = = =Linear stressLinear strain

//

σε

ππ

2

2Δ Δ,

where r is the initial radius of the wire, L is the initial length of the wire; πr 2 is the area of cross-section of the wire; M is the mass of the weights in the pan at the bot-tom due to which elongation ΔL is produced in the wire; the force applied by the mass M on the wire is equal to its weight, that is, Mg, where g is the acceleration due to gravity.

28. The ratio of tangential stress to the tangential strain pro-duced in a body within elastic limits is known as shear modulus or modulus of rigidity, which is mathemati-cally expressed as

Shear modulusShearing stressShearing strain

//

( )GF A

x LFL= = = =

σθ Δ AA xΔ

.

29. SI unit of shear modulus is N/m2 or pascals (Pa).

30. Shear modulus of elasticity exists in solids only. The value of shear modulus for a solid is less than its Young’s modulus.

31. For most of the materials, the rigidity modulus is nearly one-third of the Young’s modulus.

32. The ratio of normal stress to the volumetric strain pro-duced in the body within the elastic limits is called bulk modulus of elasticity, which is mathematically expressed as

BulkmodulusHydrostatic stress

Volume strain( )

( / )B

pV V

pV= =−

= −Δ ΔVV

,

where the negative sign shows that with increase in pres-sure p, the volume of the body decreases, that is, if p is positive, ΔV is negative. Hence, for a system in equilib-rium, the value of bulk modulus should be positive.

33. The SI unit of bulk modulus is N/m2 or pascals (Pa).

34. The reciprocal of the bulk modulus of a material is called the compressibility of that material and is rep-resented by the symbol k. Compressibility is defined as the fractional change in volume per unit increase in pressure. Mathe matically, compressibility of a material is expressed as

Compressibility ( )( / )

.kB pV V

VpV

= =−

= −1 1Δ

Δ

35. Bulk modulus for solids is greater than that for liquids which in turn is greater than the bulk modulus for gases.

36. When a deforming force is applied at the free end of a sus-pended wire, the ratio of lateral strain and the longitudi-nal strain produced in the wire is called Poisson’s ratio, which is mathematically expressed as

Poisson’s ratio ( )Lateral strain

Longitudinal strain//

σ = =−

ΔΔ

l lR RR

R ll R

= − ΔΔ

,

where l is the initial length and R is the radius of the wire before applying the deforming force and Δl and ΔR are



the increase in length and decrease in radius after the wire is stretched.

37. Theoretically, the value of Poisson’s ratio must lie between −1 and + 0.5 for all substances. However, practically it lies between 0 and +0.5.

38. If a body under tension suffers no lateral contraction, the value of Poisson’s ratio for it is zero.

39. Applications of elastic behavior of materials: When loaded at the center and supported near its ends, a bar (bridge, buildings etc.) sags by a quantity

δ = Wlbd Y

3

34,

where l is the length of the bar, b is the breadth of the bar, d is the depth of the bar, and Y is the Young’s modulus of

the material. (On increasing the depth d of a bar, unless the load is exactly at the right place, the deep bar bends, which effect is known as buckling.)

40. When a wire is stretched, the deforming force does some work against the internal restoring forces acting between the various particles of the wire. This work done is stored in the form of potential energy in the wire and is known as elastic potential energy.

41. The elastic potential energy stored in a wire is given by

U = × ×12

2Young’smodulus Strain( ) .

42. A hollow shaft is stronger than a solid shaft made of same material because the torque required to produce a given twist in a hollow shaft is greater than that required to twist a solid shaft through the same angle.

TARGET COMPETITION

1. Solids have a definite shape and size. A force is required in order to deform, change the shape or size of a body. For example, the length of a rubber band increases when we stretch the rubber band by applying some force on it. When the force applied on the rubber band is removed, the rubber band regains its original configuration. This property by virtue of which a body regains its original configuration when the deforming force is removed is known as elasticity. The deformation caused in the body is known as elastic deformation. However, some sub-stances such as a lump of putty or mud do not regain their original configuration at all after removing the deforming force and get permanently deformed. Such substances are known as plastic substances and this property is known as plasticity.

2. Elastic behavior of solids: When a large number of atoms come together to form a metallic solid, such as an iron nail, they settle into equilibrium positions in a three-dimensional lattice, a repetitive arrangement in which each atom is at a well-defined equilibrium distance from its nearest neighbors. The atoms are held together by interatomic forces that are modeled as tiny springs, as shown in Fig. 8.1a. The lattice is remarkably rigid, which is another way of saying that the “interatomic springs” are extremely stiff. It is for this reason that we perceive many ordinary objects, such as metal lad-ders, tables, and spoons, as perfectly rigid. Of course, some ordinary objects, such as garden hoses or rubber gloves, do not strike us as rigid at all. The atoms that make up these objects do not form a rigid lattice, such as that of Fig. 8.1a, but are aligned in long, flexible molecular chains, each chain being only loosely bound to its neighbors.

From an atomic viewpoint, elastic behavior has its ori-gin in the forces that atoms exert on each other, and Fig. 8.1b symbolizes these forces with the aid of springs. It is because of these atomic-level “springs” that a material tends to return to its initial shape once the forces that cause the deformation are removed.

(a)

(b)

FigurE 8.1



3. Stress: When we apply a deforming force on an elastic body, it produces a change in the configuration of the body. If the deforming force is removed, the elastic body regains its original configuration with the help of a restor-ing force which acts in a direction opposite to that of the deforming force. The restoring force per unit area acting on the body is known as stress.

If F is the magnitude of the force applied on the body and A is the area of cross-section of the body, then the magni-tude of stress is given by the relation

StressRestoring force

Area= = F

A.

The SI unit of stress is same as that of pressure, that is, N/m2 or pascal (Pa) and the dimensional formula of stress is [ ].ML T1 2− −

Figure 8.2 shows three ways in which a solid might change its dimensions when forces act on it. In Fig. 8.2a, a cylinder is stretched. In Fig. 8.2b, a cylinder is deformed by a force perpendicular to its long axis, much as we might deform a pack of cards or a book. In Fig. 8.2c, a solid object placed in a fluid under high pressure is compressed uniformly on all sides. What the three deformation types have in common is that a stress, or deforming force per unit area, produces a strain, or unit deformation. Tensile stress (associated with stretching) is illustrated in Fig. 8.2a, shearing stress in Fig. 8.2b, and hydraulic stress in Fig. 8.2c.

L + ∆LL

(a)

L

∆x

(b)

∆V

V

(c)

F

F

F

F

L + ∆LL

(a)

L

∆x

(b)

∆V

V

(c)

F

F

F

F

L + ∆LL

(a)

L

∆x

(b)

∆V

V

(c)

F

F

F

F

FigurE 8.2

4. Following are the three types of stress:

a. Normal stress: When the elastic restoring force or deforming force acts perpendicular to the cross-sectional area of the body on which deforming force is applied, the stress is called normal stress. Normal stress can be subdivided into two following categories:

Tensile stress: When there is an increase in the length or the extension of the body in the direction of the force applied, the stress set up is called tensile stress. For example, let us consider a cylinder of length L, as shown in Fig. 8.2a. If a force of magnitude F is applied normally on the cross-section of the cylinder, then due to the deforming force F the length of the cylinder increases along the direction of the force applied. Let ΔL be the increase in the length of the cylinder. The stress developed in the cylinder in this case will be tensile stress as the increase in length of the cylinder is in the same direction as the deforming force. The restoring force per unit area acting in this case on the cylinder is known as tensile stress.

Compressive stress: When there is a decrease in the length or the compression of the body due to the deforming force applied normally on the body, the stress set up is called compressive stress. For example, let us consider a cylinder, shown in Fig. 8.3, on which a compressive force of magnitude F is applied. The length L of the cylinder decreases by ΔL. The restoring force per unit area in this case on the cylinder is known as compressive stress. Tensile or compressive stress can also be termed as longitudinal stress because both produce a change in the length of the body being deformed.

F

F

∆l

l

FigurE 8.3

b. Tangential or shearing stress: If two equal and opposite forces are applied parallel to the cross-sectional area of the cylinder, as shown in Fig. 8.2b, there is a relative displacement Δx between the opposite faces of the cylinder. The restoring force per unit area developed in the body due to the applied tangential force is known as tangential stress. Mathematically, it is expressed as

Shearing stressTangential force

Area= .

c. Hydraulic stress: In Fig. 8.2c, the stress is the fluid pressure p on the object, and, as you will see in Chapter 9, pressure is the force per unit area. The strain is ΔV/V, where V is the original volume of the specimen and ΔV the absolute value of the change in volume.



The corresponding modulus, with symbol B, is called the bulk modulus of the material. The object is said to be under hydraulic compression and the pressure can be called the hydraulic stress. For this situation, we write

p BV

V= Δ

.

The bulk modulus is 2.2 × 109 N/m2 for water and 1.6 × 1011 N/m2 for steel. The pressure at the bottom of the Pacific Ocean, at its average depth of about 4000 m, is 4.0 × 107 N/m2. The fractional compression ΔV/V of a volume of water due to this pressure is 1.8%; that for a steel object is only about 0.025%. In general, solids – with their rigid atomic lattices – are less compressible than liquids, in which the atoms or molecules are less tightly coupled to their neighbors.

5. Stress and strain: When we apply a deforming force on an elastic body, it produces a change in the configuration of the body. The change in configuration per unit original configuration is known as strain.

Strain = Change in configuration

Original configuration.

Strain is a dimensionless quantity. The change in configu-ration of the body produced by the deforming force can be either change in length, volume, or shape.

6. Strain is of following three types:

a. Longitudinal strain: If the deforming force acting on an elastic body produces a change in only the length of the body, then the change in length per unit original length of the body is known as longitudinal strain. For example, in the case of the cylinder shown in Fig. 8.2a the normal stress acting on the cylinder produces a change in length ΔL in the cylinder. The longitudinal strain for the cylinder will be

Longitudinal strain=Change in length

Original length= ΔL

L.

b. Shearing strain: If the deforming force acting on an elastic body, such as the one shown in Fig. 8.4, produces a change in the shape of the body without changing its volume, then the strain produced in the body is known as shearing strain. For example, let us consider a cube in Fig. 8.4 in which the lower face ABCD is fixed and a tangential force of magnitude F acts on the upper face EFGH. The upper face shifts its position to E′F′G′H′as shown in Fig. 8.4 and the vertical faces ADFE and BCGH laterally shift to positions ADF′E′ and BCG′H′ respectively, through an angle θ. We also define shearing strain as the ratio of the displacement of a surface under a tangential force to the perpendicular distance of the displaced surface from the fixed surface. This statement can be mathematically expressed as

Shearing strain ( ) .θ = ΔLL

A

E E H H�

G�F �F G

F

B

D CL

∆L

q

q q

q

Fixed face

FigurE 8.4

c. Volumetric strain: If the deforming force acting on an elastic body produces a change in the volume of the body alone, then change in volume per unit original volume of the body is known as volumetric strain. For example, in Fig. 8.2c, where a solid sphere is subjected to a uniform hydraulic stress from a fluid, the sphere shrinks in volume by an amount ΔV. The normal hydraulic stress acting on the sphere produces a decrease in its volume. The volumetric strain for the sphere will be

Volumetric strain=Change in volume

Originalvolume= ΔV

V.

7. Hooke’s law: According to Hooke’s law, under elastic limits and for small deformations, the stress and strain are directly proportional to each other. The elastic limit is the maximum limit of deforming force up to which, if deforming force is removed, the body regains its original configuration com-pletely and if the deforming force exceeds the elastic limit, the body loses its property of elasticity and gets perma-nently deformed. This can be mathematically expressed as

Stress StrainStress Strain,

∝= ×k

where k is a constant of proportionality and is known as the modulus of elasticity. Hooke’s law is applicable to most of the materials but there are certain materials in which the relationship between stress and strain is not linear and they do not obey Hooke’s law.

8. Stress – strain curve: The relationship between stress and strain may be best studied by plotting a graph between the two for various values of stress and the accompanying strain. A typical graph for a metal is shown in Fig. 8.5. This graph is a simple case of a metallic bar or wire subjected to an increasing tension.

When the stress exceeds the elastic limit, the strain increases more rapidly than the stress, and the graph shows curve along AB, the extension of the wire now being partly elastic and partly plastic. Hence, if the wire is unloaded at point B it does not come to its original configuration along OA, but takes the dotted path BC, so that there remains a residual strain OC in the wire even when the stress is zero. We say that the wire has acquired permanent set.

If the stress on the wire is further increased beyond point B, for very little or no increase in stress, there is large but



erratic increase in strain up to point D. Thus, the portion BD of the curve in Fig. 8.5 is an irregular wavy line, the stress corresponding to D being less than that corre-sponding to B. The point B where the large increase in strain commences is called the yield point. Yield point is also sometimes known as commercial elastic limit. The stress corresponding to yield point B is known as yielding stress.

The yielding stops at point D and further extension or increase in wire length which now becomes plastic can only be produced by gradually increasing the load so that the portion DF of the stress–strain curve is obtained. In the region DF, the cross-section of the wire decreases uniformly with extension up to F and hence its volume remains constant. The maximum load to which the wire is subjected divided by its original cross-sectional area is called the ultimate strength or the tensile strength of the wire and is sometimes also termed as breaking stress.

If the load is decreased – that is, the stress is reduced – the wire finally snaps or breaks at point E which thus repre-sents the breaking point for the wire. If in the stress–strain curve for a material the ultimate strength and fracture points E and F are close, the material is said to be brittle and if they are far apart the material is said to be ductile.

Substances such as tissue of aorta, rubber, etc., which can be stretched to cause large strains are called elastomers. Ductile materials show large plastic range beyond elastic limit. The breaking point is widely separated from point of elastic limit on the stress–strain curve. Examples of ductile materials are copper, silver, iron, aluminum, etc. Brittle materials show very small plastic range beyond elastic limit. The breaking point lies close to the elastic limit. Examples of brittle materials are cast iron, glass, etc.

StrainC G x0

‘Permanentset’

Elasticrange

Elasticlimit

PlasticrangeA

B

y

D

FE

Stre

ss

FigurE 8.5

9. Elastic aftereffect is the temporary delay in regaining the original configuration by an elastic object when all the deforming forces acting on the object are removed

10. Elastic fatigue The property of an elastic body due to which its behavior becomes less elastic under the action of repeated alternating deforming forces is called elastic fatigue. For example, on stretching and releasing the rub-ber again and again, the elasticity of rubber is lost.

11. Elastic moduli: The ratio of the stress to the correspond-ing strain produced in a body within the elastic limits is

called modulus of elasticity or coefficient of elasticity. We know that according to Hooke’s law, stress is directly proportional to strain. On converting the proportiona-lity sign to an equality, a constant is introduced which is known as the modulus of elasticity. Modulus of elasticity is numerically equal to the ratio of stress and strain and, therefore, it has same dimensions as stress:

Modulus of elasticity = StressStrain

.

12. Depending upon the different types of stress and strain there are three different types of modulus of elasticity: (1) Young’s modulus of elasticity, (2) shear modulus of elasticity, and (3) bulk modulus of elasticity.

13. Young’s modulus: The ratio of normal stress to the longi-tudinal strain within the elastic limit is called Young’s mod-ulus of elasticity. When the deforming force is applied to the body only along a particular direction, the change per unit length in that direction is called longitudinal, linear, or elongation strain, and the force applied per unit area of cross-section is called the longitudinal or linear stress. The ratio of longitudinal stress to linear strain, within elas-tic limits is called Young’s modulus, and may be denoted by the symbol Y.

Y = =Tensile (or compressive) stressLinear strain

σε

.

Tensile stress is defined as the force per unit area, that is σ = ( ),F A/ and linear strain is the ratio of change in length to the original length, that is ΔL L/ .

YF A

L LFL

A L= =/

/Δ Δ.

Since strain is a dimensionless quantity and Young’s mod-ulus is the ratio of stress and strain, the dimensions of Young’s modulus will be the same as that of stress and its units will be N/m2 or pascal (Pa).

14. Determination of Young’s modulus of the material of a wire: The apparatus consists of two straight wires labeled A and B of a given material (Fig. 8.6) and having the same length and area of cross-section (or same radius). They are suspended side by side, parallel to each other, with the help of a rigid support. Wire A is called the reference wire and wire B is called the experimental wire. Wire A carries a millimeter main scale M and wire B carries a pan in which known weights can be placed. At the bottom of the experimental wire B, a vernier scale V is attached to a pointer and the main scale is fixed to the reference wire A. When weights are placed in the pan provided at the bot-tom of the experimental wire B, due to tensile stress, the length of the experimental wire increases. The increase in length of the experimental wire B can be measured using the vernier arrangement.

Let us assume that r is the initial radius and L is the initial length of the experimental wire B. The area of cross-sec-tion of the wire will be πr 2 . If M is the mass of the weights in the pan at the bottom due to which elongation ΔL is



produced in the wire, the force applied by the mass M on the wire will be equal to its weight, that is, Mg, where g is the acceleration due to gravity, the Young’s modulus of the wire can be calculated using the relation

YMg r

L LMgLr L

= = = =Linear stressLinear strain

//

σε

ππ

2

2Δ Δ.

ReferenceWire

M(Meter Scale)

01

23

45

67

V(Vernier Scale)

ExperimentalWire

BA

FigurE 8.6

15. Shear modulus: The ratio of the shearing stress to the corresponding shearing strain developed in the body is called the shear modulus of the material and is rep-resented by the symbol G. The shearing modulus is also known as modulus of rigidity. Mathematically, shearing or rigidity modulus is expressed as

Shear modulusShearing stressShearing strain

//

( )GF A

x LFL= = = =

σθ Δ AA xΔ

.

SI unit of shear modulus is N/m2 or pascals (Pa). For most of the materials, the rigidity modulus is nearly one-third of the Young’s modulus, that is, G Y= /3.

16. Bulk modulus: The ratio of the hydrostatic stress to the corresponding volume strain (or hydrostatic strain) is called bulk modulus. Bulk modulus is represented by the symbol B. Mathematically, bulk modulus is expressed as

BulkmodulusHydrostatic stress

Volume strain( )

( / )B

pV V

pV= =−

= −Δ ΔVV

.

The SI unit of bulk modulus is N/m2 or pascals (Pa). The negative sign shows that with increase in pressure p, the volume of the body decreases. That is, if p is positive, ΔV is negative. Hence, for a system in equilibrium, the value of bulk modulus should be positive.

The reciprocal of the bulk modulus of a material is called the compressibility of that material and is represented

by the symbol k. Compressibility is defined as the frac-tional change in volume per unit increase in pressure. Mathematically, compressibility of a material can be expressed as

Compressibility( )( / )

.kB pV V

VpV

= =−

= −1 1Δ

Δ

17. Poisson’s ratio: It is a commonly observed fact that when we stretch a string or a wire, it becomes longer but also thinner. Similarly, when a metallic wire is allowed to be stretched under the effect of deforming forces, the length of the wire increases, that is, longitudinal strain is produced in the wire. However, with the increase in the length of the wire, the cross-sectional area of the wire decreases. Thus, we conclude that linear or longitudinal strain produced in the wire is accompanied by a transverse or lateral strain in the wire. Finally, the radius of the wire decreases due to the lateral strain. The ratio of the lateral strain and longitu-dinal strain is called Poisson’s ratio: Poisson’s ratio is rep-resented by the symbol σ and is a dimensionless quantity. If l and R are the initial length and radius of the wire before applying the deforming force and Δl and ΔR are the increase in length and decrease in radius after the wire has been stretched, then Poisson’s ratio can be expressed as

Poisson’s ratio ( )Lateral strain

Longitudinal strain//

σ = =−

ΔΔ

l lR RR

R ll R

= − ΔΔ

.

18. Elastic potential energy in a stretched wire: When a wire is stretched, the deforming force does some work against the internal restoring forces acting between the various particles of the wire. This work done is stored in the form of potential energy in the wire and is known as elastic potential energy.

19. Elastic potential energy stored in a wire is given by

U = × ×12

2Young’s modulus Strain( ) .

20. Applications of elastic behavior of materials: While designing a building we make use of beams and columns. In the case of both bridges and buildings, the beams are designed such that they should not bend too much or break under load. To understand this in detail, let us con-sider a bar of length l, breadth b, and depth d, as shown in Fig. 8.7. When loaded at the center and supported near its ends, it sags by

δ = Wlbd Y

3

34.

d

W

l

d

FigurE 8.7


Chapter 9Mechanical Properties of Fluids

TARGET CBSE

1. Fluids include both liquids and gases. When external forces are present, a fluid, unlike a solid, can flow until it conforms to the boundaries of its container.

2. The uniform density r of a fluid is expressed as

ρ = mV

,

where m is the mass of the of the fluid and V is the volume of the fluid.

3. Density is a scalar property; its SI unit is the kilogram per cubic meter.

4. Pressure is defined as the normal force acting per unit area of a fluid, which is expressed as

P = FA

,

where F is the magnitude of the normal force on area A. (When we say a force is uniform over an area, we mean that the force is evenly distributed over every point of the area.)

5. The SI unit of pressure is the newton per square meter, which is given a special name, pascal (Pa), which is same as N/m2. Pascal is related to some other common (non-SI) pressure units as follows:

1 1 01 10 7605atm Pa torr= × =. .

6. Pascal’s law: A change in the pressure applied to an enclosed incompressible fluid is transmitted undimin-ished to every portion of the fluid and to the walls of its container.

7. The pressure in a fluid varies with depth (h) as per the expression

P = Pa + r g h,

where r is the density of the fluid, when it is uniform.

8. Hydrostatic pressure is the pressure due to fluid that is static (at rest).

9. The pressure at a point in a fluid in static equilibrium depends on the depth of that point but not on any hori-zontal dimension of the fluid or its container.

10. For an incompressible fluid passing any point every second in a pipe of non-uniform cross-section, the volume is the same in the steady flow, that is,

vA = constant,

where v is the velocity and A is the area of cross-section.

11. The equation is due to mass conservation in incompres-sible fluid flow.

12. The device used to measure the pressure of the atmo-sphere is called mercury barometer.

13. The device used to measure the gauge pressure (Pg) of a gas is called open-tube manometer.

14. Hydraulic lever works on the principle of Pascal’s law. With a hydraulic lever, a given force applied over a given distance can be transformed to a greater force applied over a smaller distance.

15. Archimedes’ principle: When a body is fully or partially submerged in a fluid, a buoyant force

Fb from the sur-rounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight mf g of the fluid that has been displaced by the body.

16. When a body floats in a fluid, the magnitude Fb of the buoyant force on the body is equal to the magnitude Fg of the gravitational force on the body.

17. When a body floats in a fluid, the magnitude Fg of the gravitational force on the body is equal to the weight mf g of the fluid that has been displaced by the body.

18. The upward force exerted on an object immersed in a fluid is equal to the weight of the fluid that the object displaces. This force exerted by the fluid on the object is called the buoyant force and the effect is known as buoyancy.

19. Depending on the nature of flow of fluids, there are four assumptions that we make about ideal fluid:

a. Steady flow: In steady (or laminar) flow, the velocity of the moving fluid at any fixed point does not change with time.

b. Incompressible flow: When the density of fluids has a constant, uniform value at rest, they show incompress-ible flow.

c. Non-viscous flow: When an object moves through a non-viscous fluid, it would experience no viscous drag force, that is, no resistive force due to viscosity of the fluid.

d. Irrotational flow: In an irrotational flow, a body does not rotate about an axis through its own center of mass.

20. Bernoulli’s principle: As we move along a streamline, the sum of the pressure (P), the potential energy per unit vol-ume (rgy), and the kinetic energy per unit volume (rv2/2) remains a constant:

P + rv2/2 + rgy= constant,

which is basically the conservation of energy applied to non-viscous fluid motion in steady state. There is no fluid that have zero viscosity and hence the above statement is treated true only approximately. The viscosity is similar to friction that converts the kinetic energy to heat energy.


Chapter 9 Mechanical Properties of Fluids42

21. If the speed of a fluid element increases as the element travels along a horizontal streamline, the pressure of the fluid must decrease, and conversely.

22. Torricelli’s law: “Efflux” means fluid outflow. Torricelli discov-ered that the speed of efflux from an open tank is expressed by a formula identical to that of a free-falling body:

v1= 2gh,

when the tank is exposed to the atmosphere, that is, P = Pa ; this equation is known as Torricelli’s law.

23. Venturi-meter is a device used to measure the speed of the flow of incompressible fluid.

24. In a fluid, though shear strain does not require shear stress, when a shear stress is applied to a fluid, the motion is gen-erated which causes a shear strain growing with time. The ratio of the shear stress to the time rate of shearing strain is called coefficient of viscosity (h).

25. The SI unit of coefficient of viscosity is poiseiulle (Pi) or N s/m2 or Pa s.

26. Force to move a layer of viscous fluid with a constant velocity: The magnitude of the tangential force

F required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular distance y from an immobile surface, is given by

FAvy

=η

,

where h is the coefficient of viscosity.

27. SI unit of viscosity is Pa s. Common unit of viscosity is poise (P).

28. Poiseuille’s law: A fluid whose viscosity is h, flowing through a pipe of radius R and length L, has a volume flow rate Q given by

QR P P

L= -π

η

42 1

8( )

,

where P1 and P2 are the pressures at the ends of the pipe.

29. According to Stokes’s law, viscous force F acting on the sphere varies directly with (1) the coefficient of viscos-ity h of the fluid, (2) velocity v of the spherical body, and (3) radius r of the spherical body. Stokes’s law – the viscous dragging force – is mathematically expressed as

F a v= 6 π η ,

which explains the retarding force which is proportional to the velocity.

30. The constant velocity, acquired by a freely falling body in a viscous medium, is known as terminal velocity.

31. Reynolds number: The onset of turbulence in a fluid is determined by a dimensionless parameter given by

Rv d

e ,= ρη

where d is a typical geometrical length associated with the fluid flow.

32. Surface tension is a property by virtue of which, the free surface of a liquid possesses a tendency to contract so as to acquire a minimum surface area. If F be the force acting and l the length of the imaginary line, then the surface tension is given by

SFI

= .

33. The SI unit of surface tension is N/m. The dimensional for-mula of surface tension is [ML0T-2].

34. For a given volume, spheres have the least surface area and so raindrops, soap bubbles, drops of mercury assume a spherical shape.

35. Surface energy: The potential energy per unit area of the surface film is called the surface energy. It is the amount of work done in increasing the area of a surface film through unity under isothermal conditions:

Surface energyWork done in increasing the surface area

Inc=

rrease in surface area.

36. The SI unit of surface energy is N/m and dimension of sur-face energy is [MT-2].

37. The angle between tangent to the liquid surface at the point of contact and solid surface, inside the liquid, is termed as angle of contact and is denoted by q.

38. A liquid whose angle of contact is less than 90° suffers capillary rise, a liquid whose angle of contact is greater than 90° suffers capillary depression and when angle of contact is q = 90°, the liquid will neither rise nor fall.

39. The liquid rises more in a narrow tube and less in a wider tube.

TARGET COMPETITION

1. A fluid, in contrast to a solid, is a substance that can flow. Fluids take up the shape of the container in which we put them because they cannot sustain a force that is tangen-tial to their surface. It can, however, exert a force in the direction perpendicular to its surface. Liquids and gases together are called fluids because there is not any orderly long-range arrangement that is found in solids.

2. The uniform density r of a fluid at any point is expressed as

ρ = mV

,

where m and V are the mass and volume of the fluid. Density is a scalar quantity; its SI unit is kilogram per cubic meter (kg/m3). The density of a gas varies considerably



with pressure, but the density of a liquid does not, that is, gases are readily compressible but liquids are not.

3. The pressure from the fluid on a surface of area ∆A is expressed as

PFA

= ∆∆

,

where ∆F is the magnitude of the force (exerted by the fluid) that acts normal to the surface. In theory, the pres-sure at any point in the fluid is the limit of this ratio as the surface area ∆A of the surface, centered on that point, is made smaller and smaller. If the force is uniform over a flat area A, pressure is given by

PFA

= ,

where F is the magnitude of the normal force on area A. This equation involves only the magnitude of force, which is a scalar quantity, hence Pressure is a scalar having no directional properties.

The SI unit of pressure is newton per square meter, which is given a special name, pascal (Pa); 1 pascal is related to some other common (non-SI) pressure units as follows:

1 1 01 10 760 14 75atm Pa torr lb/in2= × = =. . .

The unit “atmosphere (atm)” is the approximate average pressure of the atmosphere at sea level. The unit torr is formerly called the millimeter of mercury (mm Hg). The pound per square inch is often abbreviated psi.

4. Fluids at rest: In a tank of water – or other liquid – open to the atmosphere, the pressure increases with depth below the air–water interface and decreases with altitude as one ascends into the atmosphere. The pressures encountered at such extremes are usually called hydrostatic pressures because they are due to fluids that are static (at rest). Pres-sure at depth h is given by

P P gh= +0 ρ .

It is inferred from this equation that pressure at a point in a fluid in static equilibrium depends on the depth of that point but not on any horizontal dimension of the fluid or its container. In the above equation, P is said to be the total pressure or absolute pressure at level 2. The pressure P at level 2 consists of two contributions: (1) P0, the pressure due to the atmosphere that acts on the liquid and (2) rgh, the pressure due to the liquid above level 2, which acts down on level 2. In general, the difference between an absolute pressure and an atmospheric pressure is called the gauge pressure.

5. Measuring Pressure

Mercury barometer: A device used to measure the pres-sure of the atmosphere is a long glass tube filled with mercury and inverted with its open end in a dish of mercury. The space above the mercury column contains

only mercury vapor, whose pressure is so small at ordinary temperatures that it can be neglected.

Open-tube manometer: A device that measures the gauge pressure Pgof a gas. It consists of a U-tube contain-ing a liquid, with one end of the tube connected to the vessel whose gauge pressure we wish to measure and the other end open to the atmosphere.

P P P g hg ,= - =0 ρ

where r is the density of the liquid in the tube. The gauge pressure Pgis directly proportional to h. The gauge pres-sure can be positive or negative, depending on whether P > P0 or P < P0. When the gauge pressure is a positive quantity, it is sometimes called the over pressure.

6. Pascal’s principle is stated as a change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.

Consider the case in which the incompressible fluid is a liquid contained in a tall cylinder. The cylinder is fitted with a piston on which a container of lead balls rests. The atmosphere, container, and balls exert pressure Pext on the piston and thus on the liquid. The pressure P at any point P in the liquid is then

P P g h= +ext ,ρ

where h is the depth of point P from the liquid surface.

7. Pascal’s principle and hydraulic lever: The work W done on the input piston by the applied force is equal to the work W done by the output piston in lifting the load placed on it:

W F d FAA

dAA

Fd= =

=0 00

0i

ii

ii i ,

where Fi is the magnitude of external force of directed downward on the left-hand (or input) piston, whose sur-face area is Ai. An incompressible liquid in the device then produces an upward force of magnitude F0 on the right-hand (or output) piston, whose surface area is A0. Here, di is the distance moved by the input piston toward down-ward direction and d0 is the distance moved by the output piston toward upward direction.

8. The net upward force acting on a body immersed in a liq-uid is called the buoyant force Fb. Archimedes’ principle is stated as when a body is fully or partially submerged in a fluid, a buoyant force Fb from the surrounding fluid acts on the body. The force is directed upward and has a mag-nitude equal to the weight mf g of the fluid that has been displaced by the body.

The buoyant force on a body in a fluid has the magnitude

F m gb f ,=

Where mf is the mass of the fluid that is displaced by the body.


Chapter 9 Mechanical Properties of Fluids44

9. A body is said to be in static equilibrium and floating, when the magnitude Fb of the buoyant force on the body is equal to the magnitude Fg of the gravitational force on the body, that is,

F Fb g ( ).= floating

But F m gb f .= Therefore,

F m gg f . ( )= floating

So we can also say that when a body floats in a fluid, the magnitude Fg of the gravitational force on the body is equal to the weight mf g of the fluid that has been displaced by the body. In other words, a floating body displaces its own weight of fluid.

10. The apparent weight (inside a fluid) is related to the actual weight of a body and the buoyant force on the body by

Apparent weight = Actual weight – Magnitude of buoyant force,

that is,

Weight Weight .bapp = - F

This is why objects weigh less underwater. Now the mag-nitude of the buoyant force on a floating body is equal to the body’s weight.

11. The magnitude of the resultant hydrostatic force on a curved surface is given by

F F FR H V( ) ( ) ,= +2 2

where FH and FV represent the horizontal and vertical com-ponents of the force that the tank exerts on the fluid.

12. In the case of linear accelerated motion of a fluid, the pressure variation is given by the equation

dP g a dyy= +ρ( ) ,

which states that the pressure on the bottom of a liquid-filled tank which is resting on the floor of a moving eleva-tor and accelerating upward is more than that of the tank at rest (or moving with a constant velocity). For a freely falling fluid mass (ay = − g), the pressure variation in all three coordinate directions is zero, hence the pressure throughout will be the same.

13. Motion of a fluid in a rotating vessel: For a cylindrical vessel containing a fluid, rotating at constant angular velocity about its axis (Fig. 9.1a), the fluid particles also undergo circular motion like a rigid body. The accelera-tion of fluid particles located at a distance r from the axis of rotation will be equal to w 2 r, and the direction of the acceleration is toward the axis of rotation. Let us consider a small horizontal cylinder of length dr and cross-sectional area A located at a distance y below the free surface of the fluid and at a distance r from the axis. The net horizontal force acting on the cylinder is equal to the product of mass (dm) and acceleration.

dm Ad r= ρ

P A P A A d r r2 12- = ( )ρ ω

If left face of the cylinder is at a distance y below the free surface of the fluid then the right surface is y + dy below the surface of liquid. Thus,

P P g d y2 1- = ρ .

Solving, we get

dydr

rg

=ω2

2

and, therefore, the equation for surfaces of constant pres-sure is

yrg

= +ω2 2

2constant.

This equation shows that these surfaces of constant pres-sure are parabolic as shown in Fig. 9.1b.

r

w

ar = rw2

Axis ofrotation

(a)

(b)

Constantpressure

lines

z

x

y

r

P1P1

P2 P2

P3 P3

P4P4 w2r 2

2g

FIguRe 9.1

14. Buoyant force: The location of the line of action of the buoyant force can be determined by adding torques of pressure forces with respect to some convenient axis (Fig. 9.2). The buoyant force must pass through the center of mass of the displaced volume. The point through which the buoyant force acts is called the center of buoyancy. The same results apply to floating bodies which are only partially submerged if the density of the fluid above the liquid surface is very small compared with the liquid in which the body floats. Here the fluid is assumed to have a constant density.



C

Fb

Centroidof displacedvolume

(a)

(b)

C

Fb

FIguRe 9.2

15. Fluid dynamics: There are four assumptions regarding the flow of an ideal fluid:

a. Steady flow: In steady (or laminar) flow, the velocity of the moving fluid at any fixed point does not change with time, either in magnitude or in direction.

b. Incompressible flow: Ideal fluid is incompressible; that is, its density has a constant, uniform value.

c. Non-viscous flow: Viscosity of a fluid is a measure of how resistive the fluid is to flow. An object moving through a non-viscous fluid would experience no viscous drag force – that is, no resistive force due to viscosity; it could move at constant speed through the fluid.

d. Irrotational flow: Fluid flow is irrotational. In irrotational flow, a body in a fluid will not rotate about an axis through its own center of mass.

16. The relation between speed and cross-sectional area for the steady flow of an ideal fluid is called the equation of

continuity. In general, the volume ∆V of fluid with speed v that has passed through a particular cross-sectional area A in the interval ∆t is given by

∆ = ∆V Av t.

Considering both the left and the right sides we can write as

∆ ∆ ∆V A v t A v t

A v A v

= ==

1 1 2 2

1 1 2 2 (equation of continuity)

The equation of continuity therefore predicts that flow speed increases when we decrease the cross-sectional area through which the fluid flows. The above equation is written as

R Avv = = a constant (volume flow rate, equation of continuity)

in which Rv is the volume flow rate of the fluid (volume past a given point per unit time). Its SI unit is the cubic meter per second (m3/s). If the density r of the fluid is uniform, then expression for mass flow rate Rm (mass per unit time) is:

R R Avm v= = =ρ ρ a constant (mass flow rate).

The SI unit of mass flow rate is the kilogram per second (kg/s).

17. Bernoulli’s equation is given by

P v gy+ + =12

2ρ ρ a constant

Bernoulli’s equation is valid only for ideal fluids.


Chapter 10Thermal Properties of Matter

TARGET CBSE

1. Temperature is an SI base quantity which is related to our sense of hot and cold. It is measured with a thermometer, which contains a working substance with a measurable property, such as length or pressure that changes in a regular way as the substance becomes hotter or colder.

2. In the SI system, temperature is measured on the Kelvin scale, which is based on the triple point of water (273.16 K). Other temperatures are then defined by use of a constant-volume gas thermometer, in which a sample of gas is maintained at constant volume so its pressure is proportional to its temperature. We define the tempera-ture T as measured with a gas thermometer to be

TPP

=

→

( . ) lim ,273 160

3

Kgas

where T is in kelvins, and P3 and P are the pressures of the gas at 273.16 K and the measured temperature, respectively.

3. Liquid water, solid ice, and water vapor (gaseous water) can coexist, in thermal equilibrium, at only one set of val-ues of pressure and temperature, called the triple point of water. By international agreement, the triple point of water has been assigned a value of 273.16 K as the standard fixed-point temperature for the calibration of thermometers.

4. An ideal gas is an idealized model for real gases that have sufficiently low densities. The condition of low den-sity means that the molecules of the gas are so far apart that they do not interact (except during collisions that are effectively elastic). The ideal gas law expresses the relationship between the absolute pressure, the Kelvin temperature, the volume, and the number of moles of the gas, which is given by

PV RT= µ ,

where m is the number of moles and R is the universal gas constant.

5. Experiments have shown that there exists a lowest possi-ble temperature, below which no substance can be cooled. This lowest temperature is defined to be the zero point on the Kelvin scale and is referred to as absolute zero.

6. The Celsius temperature scale is expressed as

T TC . ,= − °273 15

where T is the Kelvin absolute temperature in kelvins and TC is the Celsius scale. The Fahrenheit temperature scale is expressed as

T TF C .= + °95

32

7. All objects change size with changes in temperature. For a temperature change ΔT, a change ΔL in any linear dimen-sion L is given by

∆ = ∆L L Tα ,

in which a is the coefficient of linear expansion. The change ΔV in the volume V of a solid or liquid is

∆ = ∆V V Tβ ,

where, b = 3a is the material’s coefficient of volume expansion.

8. Heat (Q) is energy that is transferred from a higher tem-perature object to a lower-temperature object because of the difference in their temperatures. It can be measured in joules (J), calories (cal), kilocalories (Cal or kcal), or Brit-ish thermal units (Btu), with

1 3 968 10 4 18683cal Btu J= × =−. . ,

The SI unit for heat is joule (J).

9. The internal energy of a substance is the sum of the kinetic, potential, and other kinds of energy that the molecules of the substance have.

10. If heat Q is absorbed by an object, the object’s tempera-ture change Tf - Ti is related to Q by

Q C T T= −( ),f i

in which C is the heat capacity of the object. If the object has mass m, then

Q cm T T= −( ),f i

where c is the specific heat of the material making up the object. The molar specific heat of a material is the heat capacity per mole, which means per 6.02 × 1023 elemen-tary units of the material.

11. For solids and liquids, the specific heats under constant pressure and constant volume differ usually by no more than a few percent. Gases have quite different values for their specific heats under constant-pressure conditions and under constant-volume conditions.

12. Heat absorbed by a material may change the material’s physical state – for example, from solid to liquid or from liquid to gas. The amount of energy required per unit mass to change the state (but not the temperature) of a particular material is its heat of transformation L. Thus,

Q Lm= .

13. The heat of vaporization LV is the amount of energy per unit mass that must be added to vaporize a liquid or


Chapter 10 Thermal Properties of Matter48

that must be removed to condense a gas. For water at its normal boiling or condensation temperature,

LV cal/ g kJ/mol kJ/kg= = =539 40 7 2256. .

14. The heat of fusion LF is the amount of energy per unit mass that must be added to melt a solid or that must be removed to freeze a liquid. For water at its normal freezing or melting temperature,

L F cal/ g kJ/mol kJ/kg= = =79 5 6 01 33. . .

15. The temperature at which the solid and the liquid states of a substance in thermal equilibrium with each other is called its melting point. It is characteristic of the sub-stance. It also depends on pressure. The melting point of a substance at standard atmospheric pressure is called its normal melting point.

16. To measure specific heat of a material, we heat a sample to some known temperature Tm, and keep it in a vessel containing water of known mass and temperature Tw , (Tw < T). We then measure the temperature of the water after equilibrium has been reached. This method is called calorimetry, and vessel in which this energy transfer occurs is called calorimeter.

17. The rate Pcond at which energy is conducted through a slab for which one face is maintained at the higher tem-perature TH and the other face is maintained at the lower temperature TC is

PQt

kAT T

LcondH C ,= = −

where each face of the slab has area A, the length of the slab (the distance between the faces) is L, and k is the thermal conductivity of the material.

18. Convection occurs when temperature differences cause an energy transfer by motion within a fluid.

19. Radiation is an energy transfer via the emission of elec-tromagnetic energy. The rate Prad at which an object emits energy via thermal radiation is given by the Stefan – Boltzmann law of radiation,

P ATrad ,= σ ε 4

where s (= 5.6704 × 10-8 W/m2 K4) is the Stefan – Boltzmann constant, e is the emissivity of the object’s surface, A is its surface area, and T is its surface temperature (in kelvins). The rate Pabs at which an object absorbs energy via ther-mal radiation from its environment, which is at the uni-form temperature Tenv (in kelvins), is

P ATabs env .= σ ε 4

20. If T1 is the temperature of the surroundings, and T2 is the temperature of the body, Newton’s Law of Cooling is stated as the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings:

∆∆

Qt

k T T= − × −( ),2 1

where k is a positive constant depending upon the area and nature of the surface of the body, T2 is the tempera -ture of the body, and T1 is the temperature of the sur-rounding medium. The plot between the temperature of the body and time is known as the cooling curve.

TARGET COMPETITION

1. On the Celsius temperature scale, there are 100 equal divi-sions between the ice point (0 °C) and the steam point (100 °C). On the Fahrenheit temperature scale, there are 180 equal divisions between the ice point (32 °F) and the steam point (212 °F). One kelvin (1 K) is equal in size to one Celsius degree.

2. Temperature conversions:

From C to F : ;

From F to C : ;

From C to kelv

F C

C F

° °

° °

°

T T

T T

= + °

= − °

95

32

59

32

iin : . .K CT T= + °273 15

3. For an object held rigidly in place, a thermal stress can occur when the object attempts to expand or contract. The stress can be large, even for small temperature changes.

4. When the temperature changes, a hole in a plate of solid material expands or contracts as if the hole is filled with the surrounding material. Similarly, a cavity in a piece of solid material expands or contracts thermally as if the cavity were filled with the surrounding material.

5. When materials are placed in thermal contact within a perfectly insulated container, the principle of energy conservation requires that heat lost by warmer materials equals to the heat gained by cooler materials.

6. Heat is sometimes measured with a unit called the kilo-calorie (kcal). The conversion factor between kilocalo-ries and joules is known as the mechanical equivalent of heat:

1 kcal = 4186 J.

7. A fusion curve for a material gives the combinations of temperature and pressure for equilibrium between the solid and liquid phases of that material.

8. The relative humidity is defined as

Percentrelative

humidity

Partial pressureof water v

=

aaporEquilibrium vapor pressure of

water at the exist

iing temperature

×100

9. The dew point is the temperature below which the water vapor in the air condenses. On the vaporization curve of



water, the dew point is the temperature that corresponds to the actual pressure of water vapor in the air.

10. The length L0 of an object changes by an amount ΔL when its temperature changes by an amount ΔT

∆ ∆L L T= α 0 ,

where a is the coefficient of linear expansion. The common unit for the coefficient of linear expansion is 1/C° = (C°)-1.

11. A bimetallic strip is made from two thin strips of metal that have different coefficients of linear expansion, a1 and a2. The radius of curvature of such a strip of thickness d when it bends due to an increase in temperature Δt is given by

Rd t

t= + + ∆

− ∆

( ( )(( ) )

.2

21 2

1 2

α αα α

12. The coefficient of superficial expansion of a substance represents the alteration in area of a sheet of the substance initially possessing unit area, when it is heated through unit difference of temperature. The coefficient of superfi-cial expansion is twice as much as the coefficient of linear expansion:

b = 2a.

13. The volume V0 of an object changes by an amount ΔV when its temperature changes by an amount ΔT is given by

∆ ∆V V T= γ 0 ,

where g is the coefficient of volume expansion. The coef-ficients of volume expansion and linear expansion for a solid are related by g = 3a. Though it is a characteristic of the substance, g is not strictly a constant. It depends in general on temperature; g becomes constant only at a high temperature. For an ideal gas, the coefficient of vol-ume expansion at constant pressure depends inversely on the temperature.

14. The way in which the level of a liquid in a container changes with temperature depends on the change in vol-ume of both the liquid and the container.

15. Since the volume of a liquid increases when heated, with-out change in mass, the density of the liquid will change on heating. If D0 is the original density of the liquid, the new density D for a temperature change ΔT is given by

D D T= − ∆0 1( ).γ

16. Above about 4 °C, water expands as the temperature rises, as we would expect. Between 0 and about 4 °C, however, water contracts with increasing temperature. Thus, at about 4 °C, the density of water passes through a maximum. At all other temperatures, the density of water is less than this maximum value.

17. The heat capacity C of an object is the proportionality constant between the heat Q that the object absorbs or loses and the resulting temperature change ΔT of the object; that is,

Q C T C T T= = −∆ ( ),f i

in which Ti and Tf are the initial and final temperatures of the object. Heat capacity C has the unit of energy per degree or energy per kelvin.

18. It is convenient to define a “heat capacity per unit mass” or specific heat c that refers not to an object but to a unit mass of the material of which the object is made. Thus, we can write

Q cm T cm T Tf i= = −∆ ( ).

c = =1 1cal/ g C Btu/ F = 4186.8 J/kg K.° °lb

When quantities are expressed in moles, specific heats must also involve moles (rather than a mass unit); they are then called molar specific heats.

19. The heat Q that must be supplied or removed to change the phase of a mass m of a substance is

Q mL= ,

where L is the latent heat of the substance. The SI unit of latent heat is J/kg.

20. The latent heat of fusion Lf refers to the change between solid and liquid phases, the latent heat of vaporization Lv applies to the change between liquid and gas phases, and the latent heat of sublimation Ls refers to the change between solid and gas phases.

21. The heat Q conducted during a time t through a bar of length L and cross-sectional area A is

QkA T t

L= ( )

,∆

where ΔT is the temperature difference between the ends of the bar and k is the thermal conductivity of the material. The conduction rate Pcond is given by

PQt

kAT T

LkA

Txcond

H C ,= = −

=

∆∆

which is known as Fourier’s law. The SI unit for thermal conductivity is J/s m K or W/m K.

22. For conduction through a composite slab, we have the expression

PA T T

L kcondH C( )( / )

.= −∑

23. The thermal resistance of a slab of thickness L is defined as

RLk

=

24. The conduction rate through a cylinder of inside radius ri , outside radius r0 , and length L, with its environment at a constant temperature Tenv , is given by

PkL T T

r r= −2π ( )

ln( / ).i env

o i


Chapter 10 Thermal Properties of Matter50

25. When heat is conducted through a multi-layered material, and the high and low temperatures are constant, the heat conducted through each layer is the same.

26. Since absorption and emission are balanced, a mate-rial that is a good absorber, like lampblack, is also a good emitter, and a material that is a poor absorber, like polished silver, is also a poor emitter. A perfect blackbody, being a perfect absorber, is also a perfect emitter.

27. The radiant energy Q, emitted in a time t by an object that has a Kelvin temperature T, a surface area A, and an emis-sivity e, is given by

Q e T At= σ 4 ,

where s is the Stefan – Boltzmann constant and has a value of 5.67 × 10-8 J / (s m2 K4).

28. A liquid which is in equilibrium with vapor has the same pressure and temperature throughout the system – the two phases in equilibrium differ in their molar volume (i.e., density). This is true for a system with any number of phases in equilibrium.

29. Heat transfer involves temperature difference between two systems (or two parts of the same system). An energy transfer that does not involve temperature difference in some way is not considered as heat. Convection involves flow of matter within a fluid due to unequal temperatures of its parts. For example, a hot bar placed under a running tap loses heat by conduction between the surface of the bar and water and not by convection within water.

30. If we denote by a, r, and t the fractions of radiant energy absorbed, reflected, and transmitted, respectively, by a body, we have

a + r + t = 1.

31. For a good absorber, a > (r + t), and for a near perfect absorber, a >> (r + t); for a good reflector, r > (a + t), and for a near perfect reflector, r >> (a + t); for a good transmitter, t > (a + r), and for a near perfect transmitter t >> (a + r).

32. If T1 is the temperature of the surroundings, and T2 is the temperature of the body, Newton’s Law of Cooling is stated as the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings:

∆∆

Qt

k T T= − × −( ),2 1

where k is a positive constant depending upon the area and nature of the surface of the body, T2 is the temper-ature of the body, and T1 is the temperature of the sur-rounding medium. The plot between the temperature of the body and time is known as the cooling curve.

33. Wien’s displacement law is given by

λmax .T = × −2 88 10 3 m K,

where lmax is the wavelength at which the curve peaks and T is the absolute temperature of the surface of the object emitting the radiation. The wavelength at the curve’s peak is inversely proportional to the absolute temperature; that is, as the temperature increases, the peak is “displaced” to shorter wavelengths.


Chapter 11Thermodynamics

TARGET CBSE

1. The branch of physics, which deals with the concepts of (1) heat, (2) temperature, and (3) inter-conversion of heat and other forms of energy, is known as thermodynamics.

2. In thermodynamics, the state of a gas is specified by measurable variables such as (1) volume, temperature, pressure, mass, and composition (which are sensed by us) and (2) entropy and enthalpy(which are not so obvious to our senses).

3. When a gas inside a closed and rigid container is comp-letely insulated from its surroundings, with fixed values of temperature, pressure, volume, mass and composition which do not change with time, is said to be in a state of thermodynamic equilibrium.

4. When two systems are kept close and are separated by an insulating (movable) wall – which does not allow flow of energy (heat) from one system to another system – is called adiabatic wall. Conversely, when two sys-tems are kept close and are separated by an insulating (movable) wall – which allows flow of energy (heat) from one system to another system – is called diathermic wall.

5. Zeroth law of thermodynamics: When two systems A and B are in thermal equilibrium with a third system C, then they are separately in thermal equilibrium with each other.

6. Heat flows from the body at higher temperature to the body at lower temperature and when the heat flow stops, the two bodies are said to be in thermal equilibrium.

7. Any bulk system consists of a large number of molecules in which the internal energy is a concept that is the sum of the kinetic energies and potential energies of these molecules. It does not include the total kinetic energy of the system.

8. Heat is the transfer of energy that happens due to the temperature difference between a system and its surroundings.

9. Work (in thermodynamics) is the transfer of energy brought about by additional means (e.g., by raising or lowering a weight connected to a piston of a cylinder containing gas, work is said to be done).

10. First law of thermodynamics is the common law of conservation of energy applied to any system in which the energy transfer from the surroundings, or to the sur-roundings, (through heat and work) is taken into account. It states that

ΔQ = ΔU + ΔW,

where ΔQ is the heat supplied to the system, ΔW is the work done by the system, and ΔU is the change in internal energy of the system.

11. Specific heat capacity of a substance is expressed as

sm

QT

= ΔΔ

1

where m is the mass of the substance and ΔQ is the heat required to change its temperature by an amount ΔT.

12. The molar specific heat capacity of a substance is exp-ressed as

CQT

= ×1µ

ΔΔ

,

where m. is the number of moles of the substance for a solid, the law of equipartition of energy gives C = 3R (where R is the universal gas constant), which agrees with the experiment at ordinary temperatures.

13. Calorie is the old unit of heat. One calorie is the amount of heat required to increase the temperature of 1 g of water from 14.5 °C to 15.5 °C; 1 cal = 4.186 J.

14. When CP and CV are molar specific heat capacities of an ideal gas at constant pressure and volume, the simple equation for the ideal gas is expressed as

CP and CV = R,

where R is the universal gas constant.

15. The relation between the state variables is called the equation of state. For an ideal gas, the equation of state is expressed as

PV = mRT,

where m. is the number of moles of the substance, R is the universal gas constant, and P, V, and T are the state variables.

16. An infinitely slow process, in which a system remains in thermal and mechanical equilibrium with the sur-roundings throughout, is called quasi-static process. In this process, the pressure and temperature of the environment can differ from those of the system only infinitesimally.

17. At temperature T, in an isothermal expansion of an ideal gas from volume V1 to V2, the heat absorbed (Q) is equal to the work done (W) by the gas, which is expressed as

Q W R TVV

= =

µ ln .2

1

18. For an adiabatic process, that is, a system which is insulated from the surroundings and heat absorbed or released is zero, we have

PV γ = constant,


Chapter 11 Thermodynamics52

where

γ = CC

P

V

.

19. In an isothermal process, the temperature (T ) of the system is kept fixed throughout the process.

20. In an isochoric process, the volume (V) remains constant, that is, no work is done on or by the gas.

21. In an isobaric process, the pressure (P) is fixed, in which case, the work done by the gas is expressed as

W P V V R T T= −( ) = −2 1 2 1µ ( ).

22. When a system returns to its initial state, the process involved is called cyclic process.

23. A device by which a system is made to undergo a cyclic process that results in conversion of heat to work is called heat engine.

24. The efficiency (h) of a heat engine is defined by

η = WQ1

,

where W is the work done on the environment in on complete cycle and Q1 is the heat input, that is, the heat absorbed by the system in one complete cycle. According to the first law of thermodynamics, for one complete cycle,

W = Q1 – Q2,

therefore,

η = −1 2

1

QQ

.

25. The substance that is accountable for doing work is called working substance, for example, steam in a steam engine is the working substance.

26. In a refrigerator (or a heat pump), the system extracts heat Q2 from the cold reservoir and discharges Q1 amount of heat to the hot reservoir, with work (W) done on the system. The coefficient of performance of a refrigerator is expressed as

α = =−

QW

QQ Q

2 2

1 2

.

27. The second law of thermodynamics over-rules some proc-esses consistent with the first law of thermodynamics, which defines the following two significant statements:

a. Kelvin – Planck statement: The process whose sole result is the absorption of heat from a reservoir and complete conversion of the heat into work is not possible.

b. Clausius statement: The process whose sole result is the transfer of heat from a colder object to a hotter object is not possible.

The second law of thermodynamics implies that no heat engine can have efficiency (h) equal to 1 or no refrigerator can have coefficient of performance (a) equal to ∞.

28. Reversible process: A process is said to be reversible if it can be reversed in a way that both the (1) system and (2) surroundings return to their original states, with no other change anywhere else in the universe.

29. Reversible process: Spontaneous processes of nature are said to be irreversible process. The idealized reversible process is a quasi-static process with no dissipative factors such as friction and viscosity.

30. Carnot engine: A reversible engine operating between two temperatures – (1) source temperature (T1) and (2) sink temperature (T2) – is called Carnot engine, which consists of two isothermal processes connected by two adiabatic processes. The efficiency of a Carnot engine is given by

η = −1 2

1

TT

,

31. No engine operating between two temperatures can have efficiency greater than that of a Carnot engine.

32. Rules governing thermo dynamical systems: If (1) Q > 0, heat is added to the system, (2) Q < 0, heat is removed to the system, (3) W > 0, work is done by the system, (4) W < 0, Work is done on the system.

TARGET COMPETITION

1. A system is a portion of matter which is separated from the rest of the universe with an insulated surface, which can consist of one or more substances.

2. Types of system: (1) Isolated system is a system that exchanges neither energy nor matter with its surround-ings. (2) Closed system is a system that can exchange energy, but not matter, with its surroundings. (3) Open system is a system that can exchange matter as well as energy with its surroundings is said to be an open system.

3. The properties associated with a macroscopic system (i.e., a system consisting of large number of particles) are called macroscopic properties. These properties are

pressure (P), volume (V ), temperature (T ), composition, density (r), etc.

4. An extensive property of a system is one which depends upon the amount of the substance present in the system such as mass (M ), volume (V ), and energy (E ).

5. An intensive property of a system is that which is inde-pendent of the amount of the substance present in the system such as temperature (T ), pressure (P), etc.

6. State of a system: A system is said to be in definite state when macroscopic properties of a system have definite val-ues. When there is a change in any one of the macroscopic properties, the system is said to change to a different state. The state of a system is fixed by its macroscopic properties.



7. As the state of a system changes with change in any of the macroscopic properties, these macroscopic properties are called state variables. Some common state variables are internal pressure, temperature, volume, energy, enthalpy, entropy, free energy, etc.

8. A system in which the macroscopic properties do not experience any change with time is said to be in thermo-dynamic equilibrium.

9. Zeroth law of thermodynamics: When two systems A and B are in thermal equilibrium with a third system C, then they are separately in thermal equilibrium with each other.

10. The manner by which a thermodynamical system changes from one state to another is called a thermodynami-cal process. Major types of thermodynamical processes (whose graphs are shown in Fig. 11.1) are as follows: (1) When the temperature (T) of a system remains con-stant throughout, the resultant process is called isother-mal process; (2) When the heat enters or leaves a system at any given step of the process involved in a system is called adiabatic process; (3) When the pressure (P) of a system remains constant during each step of the process, the process is termed as isobaric process; (4) When the volume of the system remains constant during each step of the process, the resultant process is said to be isochoric process.

Volume

Isobaric

Isothermal

Adiabatic

Pres

sure

Isoc

hor

ic

FIguRe 11.1

11. A process which is carried out infinitesimally slowly in such a manner that the system remains almost in a state of equilibrium throughout the process is called reversible process. Converesely, when a process does not take place infinitesimally slowly is called an irreversible process. Naturally occurring processes are irreversible processes.

12. Substances are generally related with a definite amount of energy that depends both on its chemical nature and its temperature, pressure, and volume, which energy is known as internal energy and it is possessed by all its con-stituent molecules. Since the value of the internal energy depends only on the state of the substance and does not depend on how that state is achieved, the absolute value of internal energy of a substance cannot be determined.

13. First law of thermodynamics states that energy can nei-ther be created nor destroyed but can be transformed from one form to another form. This is also known as the law of conservation of energy. It (otherwise called the change in internal energy) is mathematically expressed as

ΔU = Q + W,

where Q is the quantity of heat transferred from the sur-roundings to the system and W is the work done in the process. If work is done by the surroundings on the system, W is positive so that ΔU = Q + W and if the work is done by the system on the surroundings, W is negative so that ΔU = Q – W.

14. Enthalpy (H) of a system represents the total energy stored in the system:

H = U + PV.

Similar to internal energy, enthalpy is also both an exten-sive property and a state function. Hence, the absolute value of enthalpy cannot be determined, however, the change in enthalpy can be experimentally determined:

ΔH = ΔU + Δ(PV).

15. In an isothermal expansion, the work done by the gas is equal to amount of heat absorbed and

W = –nRT × lnVV

2

1

= –nRT × nPP

1

2

.

16. Second law of thermodynamics First law of thermody-namics does not indicate any information about the spon-taneity or feasibility of the process, for example, it does not indicate whether heat can flow from a cold end to a hot end or vice versa. The second law of thermodynamics sates two significant statements as follows:

a. Kelvin – Planck statement: The process whose sole result is the absorption of heat from a reservoir and complete conversion of the heat into work is not possible.

b. Clausius statement: The process whose sole result is the transfer of heat from a colder object to a hotter object is not possible.

17. Reversible process: A process is said to be reversible if it can be reversed in a way that both the (1) system and (2) surroundings return to their original states, with no other change anywhere else in the universe.

18. Reversible process: Spontaneous processes of nature are said to be irreversible process. The idealized reversible process is a quasi-static process with no dissipative factors such as friction and viscosity.

19. Carnot engine: A reversible engine operating between two temperatures – (1) source temperature (T1) and (2) sink temperature (T2) – is called Carnot engine, which consists of two isothermal processes connected by two adiabatic processes. The efficiency of a Carnot engine is given by

η = −1 2

1

TT

,

20. No engine operating between two temperatures can have efficiency greater than that of a Carnot engine.

21. Rules governing thermo dynamical systems: If (1) Q > 0, heat is added to the system, (2) Q < 0, heat is removed to the system, (3) W > 0, work is done by the system, (4) W< 0, Work is done on the system.


Chapter 12Kinetic Theory

TARGET CBSE

1. Newton, Boyle, and many other scientists tried to describe the behavior of gases by considering that gases are made up of very small atomic particles (the size of an atom is about 1 Å = 10–10 m).

2. The behavior of gases which is explained on the basis of the idea that the gas consists of rapidly moving atoms or molecules is called kinetic theory.

3. In solids, the atoms are tightly packed, which are located at a distance of few angstroms (≈2 Å) apart.

4. Atoms belonging to one element are alike but differ from those of other elements. Group of atoms of an element combine to form the molecule of the compound.

5. In liquids, although the distance between the atoms is also approximately 2 Å, the atoms in liquids are not as strongly fixed as in solids, and can move around. This is the reason that the liquids flow.

6. In gases, atoms are located at a distance of tens of angstroms.

7. The average distance traveled by a molecule without colliding with other molecule is called mean free path.

8. Boyle’s law: At constant temperature, the volume of a given mass of gas is inversely proportional to pressure, which is expressed as

VP

∝ 1.

9. Charle’s law: When pressure of a gas is constant, the volume of a given mass of gas is directly proportional to its absolute temperature, which is expressed as

VT

= constant.

10. Ideal gas equation is given by

PV = mRT = kBNT,

where m is the number of moles, N is the number of molecules, and R = 8.314 J/mol/K and kB = R/NA = 1.38 × 10–23 J/K. The ideal gas equation is satisfied by real gases only approximately that too at low pressures and high temperatures. The gas that follows the ideal gas equation at all possible pressures and volumes is called ideal gas.

11. When a mixture of noninteracting ideal gases with m1 moles of gas 1, m2 of gas 2, and so on, are kept in an enclosed area with volume V, temperature T, and pressure P, the equation of state of mixture is given by

PV = (m1+ m2)RT or P = m1RT/V + m2RT/V + … = P1 + P2 + …,

where P1 = m1RT/V is the pressure the gas 1 that would exert at same V and T if no other gases were present in the enclosure. This phenomenon is called Dalton’s partial

pressures, that is, the total pressure of mixture of different ideal gases is equal to the sum of partial pressures of individual gases of which mixture is made of.

12. For an ideal gas, the relation of kinetic theory is given by

P nmv= 13

2 ,

where n is the density of molecules, m the mass of the molecule, and v 2 is the mean of squared speed. Along with the ideal gas equation, this kinetic theory equation provides the kinetic interpretation of temperature as follows:

12

32

32 21 2

mv k T v vk Tm

= =

=B rms

/B, .

This implies that the temperature of a gas is an amount of the average kinetic energy of a molecule, which is independent of the nature of the gas or molecule. At a fixed temperature, in a mixture of the gases, heavier molecule has the lower average speed.

13. Law of equipartition of energy is stated as follows: When a system is in equilibrium at absolute temperature T, the total energy is distributed equally in different energy modes of absorption, the energy in each mode being equal to (1/2)kBT. Each translational and rotational degree of freedom corresponds to one energy mode of absorption and has energy (1/2)kBT. Each vibrational frequency has two modes of energy (kinetic energy and potential energy) with corresponding energy equal to 2 × =1/2 k T k TB B .

14. The molar specific heat of gases is determined using the law of equipartition of energy and the resultant values are in agreement with the experimental values of specific heats of several gases. The agreement can be improved by including vibrational modes of motion.

15. Translational kinetic energy of the molecules in a gas is given by

E k NTB= 32

,

which leads to the following equation:

PV E= 23

.

16. The mean free path l is the average distance covered by a molecule between two successive collisions :

l = 12 2n dπ

,

where n is the density and d is the diameter of the molecule.


Chapter 12 Kinetic Theory56

1. Pressure and volume of a gas can be related as

PV = nRT,

which is called ideal gas equation in which n is number of moles of gas, R = NAkB is the gas constant, and T is the absolute temperature.

2. The number of moles of a gas is given by

nmM

NN

= =A

,

where m is the mass of gas containing N molecules, M is the molar mass, NA is Avagadro’s number (= 6.022 × 1023). Pressure of a gas can also be expressed as P = rRT/M, where r is the mass density of the gas.

3. Molecular nature of matter: (1) Molecules are made up of one or more atoms that constitute matter; (2) In solids, atoms and molecules are strongly attached to each other since distance between them is very less (of the order of few angstroms) and hence they cannot move; (3) In liquids, although the distance between the atoms is also approximately 2 Å, the atoms are not as strongly fixed as in solids, and can move around, which is the reason that the liquids flow; (4) In gases, atoms are located at a distance of tens of angstroms and hence they are free to move without colliding each other.

4. Boyle’s law: At constant temperature, the volume of a given mass of gas is inversely proportional to pressure, which is expressed as

VP

∝ 1.

5. Charle’s law: When pressure of a gas is constant, the volume of a given mass of gas is directly proportional to its absolute temperature, which is expressed as

VT

= constant.

6. When a mixture of noninteracting ideal gases with m1 moles of gas 1, m2 of gas 2, and so on, are kept in an enclosed area with volume V, temperature T, and pressure P, the equation of state of mixture is given by

PV = (m1+ m2)RT or P = m1RT/V + m2RT/V + … = P1 + P2 + …,

where P1 = m1RT/V is the pressure the gas 1 that would exert at same V and T if no other gases were present in the enclosure. This phenomenon is called Dalton’s partial pressures, that is, the total pressure of mixture of different ideal gases is equal to the sum of partial pressures of individual gases of which mixture is made of.

7. Kinetic theory of an ideal gas explains the following: (1)A gas is composed of large number of very small, invisible particles known as molecules, which are in state of motion with varying velocities in all possible directions; (2) Molecules travel in straight line between any two collisions; (3) The time of collision is negligible as compared with the time taken to travel the path; (4) Size of molecule

is infinitely small compared to the average distance traveled by the molecules between any two consecutive collisions; (5) Molecules exert force on each other except when they collide and all of their molecular energies are of kinetic energies; (6) Intermolecular distance in gas is much larger than that of solids and liquids and the molecules of gas are free to move in the space available to them.

8. Pressure is force per unit area so

P = F/L2 = (M/3L3)(Σ(v1)2/N),

where M is the total mass of the gas. When r is the density of gas, then

P = rΣ(v1)2/3N,

since Σ(v1)2/N is the average of squared speeds and is

written as v 2 , which is known as mean square speed. Thus, vrms=√(Σ(v1)

2/N) is known as the root mean squared speed (or rms speed) and v 2 = (vrms)

2. Thus pressure becomes

P = (1/3)r v 2 or PV = (1/3)Nm v 2 ,

Therefore, rms speed is given as vrms = √(3P/r) = √(3PV/M).

9. An ideal gas, the relation of kinetic theory is given by

P nmv= 13

2 ,

where n is the density of molecules, m the mass of the molecule, and v 2 is the mean of squared speed. Along with the ideal gas equation, this kinetic theory equation provides the kinetic interpretation of temperature as follows:

12

32

32 21 2

mv k T v vk Tm

= =

=B rms

/B, ,

This implies that the temperature of a gas is an amount of the average kinetic energy of a molecule, which is independent of the nature of the gas or molecule. At a fixed temperature, in a mixture of the gases, heavier molecule has the lower average speed.

10. Law of equipartition of energy is stated as follows: When a system is in equilibrium at absolute temperature T, the total energy is distributed equally in different energy modes of absorption, the energy in each mode being equal to (1/2)kBT. Each translational and rotational degree of freedom corresponds to one energy mode of absorption and has energy (1/2)kBT. Each vibrational frequency has two modes of energy (kinetic energy and potential energy) with corresponding energy equal to 2 × =1/2 k T k TB B .

11. Specific heat capacity of monoatomic gases: Monoatomic gas molecules have three translational degrees of freedom. From the law of equipartition of energy, average energy of an molecule at temperature T is (3/2)kBT. For a monoatomic gas, the ratio of specific heats is given by

TARGET COMPETITION



CP /CV = 5/3,

where CV is the molar specific heat capacity at constant volume and CP is the molar specific heat capacity at constant pressure.

12. Specific heat capacity of diatomic gases: A diatomic gas molecule is treated as a rigid rotator like dumbbell and has five degrees of freedom out of which three degrees of freedom are translational and two degrees of freedom are rotational. For a diatomic gas, the ratio of specific heats is given by

CP /CV = 9/7.

13. Specific heat capacity of solids: From the law of equipartition of energy, we can also determine the specific heats of solids. At constant pressure,

ΔQ = ΔU + PΔV = ΔU since for solids ΔV

is negligible and hence

C = ΔQ/ΔT = ΔU/ΔT = 3R,

which is called Dulong – Petit law. Specific heats of substance approaches to zero as T → 0.

14. Mean free path: On the basis of kinetic theory of gases, it is assumed that the molecules of a gas are continuously colliding against each other. Molecules move in straight line with constant speeds between two successive collisions and hence the path of a single molecule is a series of zigzag paths of different lengths. These paths of different lengths are called free paths of the molecule. If v is the distance traversed by molecule in one second, the mean free path is given by

l = Total distance traveled in 1 s/number of collision suffered by the molecules

= v/p σ2vn,

= 1/p σ2n.

This expression is valid only when we assume that all the molecules are at rest except the one which is colliding with the others.


Chapter 13Oscillations

TARGET CBSE

1. When a motion repeats itself at regular intervals of time, the motion is called periodic motion.

2. The smallest interval of time after which the motion is repeated is called the period (T ) of the motion.

3. The SI unit of period is second (s).

4. When a body is left at rest, it continues to remain at rest forever; however, if the body is given a small displacement from the position, a force comes into play which tries to bring the body back to the equilibrium point, by produc-ing oscillations or vibrations.

5. The total number of repetitions that occur per unit time of a periodic motion is represented by the reciprocal of its period T, which is represented by the symbol n. This quan-tity is called the frequency of the periodic motion:

ν = 1T

.

6. The unit of frequency is s–1 or Hertz (Hz).

7. Simple harmonic motion (SHM) is the projection of uni-form circular motion on the diameter of a circle in which the circular motion occurs. There are no physical exam-ples which can be considered as absolutely pure simple harmonic motion; however, examples such as the motion of a simple pendulum, for a small angular displacement, can be said as SHM.

8. The period of SHM does not depend on the amplitude or the energy or the phase constant.

9. In a simple harmonic motion, the displacement x(t) of a particle from its equilibrium position is expressed as

x t A t( ) cos( )= +ω φ

where A is the amplitude of the displacement, the quan-tity ( )ω φt + is the phase of the motion, and φ is the phase constant. The angular frequency w is related to the period and frequency of the motion by

ω π πν= =22

T.

10. The SI unit of angular frequency is radians (rad).

11. As functions of time, the particle velocity [v(t)] and accel-eration [a(t)] during SHM, respectively, are expressed as

v t A t

t A t t x t

( ) sin( );

( ) cos( ) ( ) ( ).

= − += − + ⇒ = −

ω ω φα ω ω φ α ω2 2

It is shown that both velocity and acceleration of a body executing SHM motion are periodic functions, having the

velocity amplitude v Am = ω and acceleration amplitudea Am ,= ω2 respectively.

12. The force acting on an SHM is proportional to the displace-ment and is always directed toward the center of motion.

13. At any time, a particle executing SHM has potential energy U = (1/2)kx2 and kinetic energy K = (1/2)mv2. If no fric-tion exists, the mechanical energy of the system, that is, E = K + U always remains constant despite the fact that K and U change with time.

14. A particle of mass m oscillating under the influence of a Hooke’s law restoring force given by

F = –kx,

where k is the force constant, exhibits simple harmonic motion with the following angular frequency (w) and period (T ):

ω

π

=

=

km

Tmk

;

.2

15. The unit of force constant is N/m and its dimension is [MT –2].

16. The motion of a simple pendulum moving back and forth through small angles is considered approximately to be sim-ple harmonic and its period of oscillation is expressed as

Tlg

= 2π .

17. The motion of a simple pendulum, which swings in air, dies out ultimately because both the (1) air drag and (2) the friction at the support oppose the motion of the pendulum and dissipate its energy gradually. Hence, a real pendulum is said to execute damped oscillations.

18. In damped oscillations, although the energy of the system is continuously dissipated, the oscillations remain appar-ently periodic.

19. For a SHM, when the damping force is given by

Fd = –bv,

where v is the velocity of the oscillator and b is a damp-ing constant, then the displacement of the oscillator is expressed as

x t A bt m( ) e cos( ’ ),/= +− 2 ω φ

where the angular frequency of the damped oscillator, w′, is expressed as


Chapter 13 Oscillations60

ω ’ .= −km

bm

2

24

When the damping constant is small, then ω ω’ ,≈ where w is the angular frequency of the undamped oscillator. The mechanical energy (E ) of the damped oscillator is expressed as

E t kA bt m( ) e ./= −12

2

20. When an external force with angular frequency, wd , acts on an oscillating system with natural angular frequency, w, the system oscillates with angular frequency wd . The amplitude of oscillations is the highest when wd = w , which is a condition known as resonance of the oscillation.

TARGET COMPETITION

1. When a particle moves in a way that it repeats its path regularly after equal intervals of time, its motion is said to be periodic motion.

2. The time interval required to complete one complete cycle of motion is called time period of motion.

3. If a body in periodic motion moves back and forth over the same path, the motion can be said to be vibratory or oscillatory (e.g., back and forth motion of pendulum, motion of a mass attached to a spring, vibrations of a tun-ing fork, and so on).

4. All oscillatory motions are periodic but all periodic motions are not oscillatory (e.g., motion of the Earth around the Sun is periodic but not oscillatory).

5. Simple harmonic motion (SHM) arises when force on oscillating body is directly proportional to the displace-ment from its equilibrium position and at any point of motion, this force is directed toward the equilibrium posi-tion, which is given by

F = – kx,

where k is the force constant. The negative sign indi-cated that the force opposes increase in the value of the displacement, x. SHM is the simplest form of oscillatory motion available. This force is known as the restoring force which takes the particle back toward the equilib-rium position, and opposes the increase in displacement.

6. The SI unit of force constant is N/m and its magnitude depends on elastic properties of the system.

7. Equation of a SHM: For a simple harmonic motion, the displacement x(t) of a particle from its equilibrium posi-tion is expressed as

x t A t( ) cos ( )= +ω φ

Where A is the amplitude of the displacement, the quan-tity ( )ω φt + is the phase of the motion, and φ is the phase constant. The angular frequency w is related to the period and frequency of the motion by

ω π πν= =22

T.

Figure 13.1 shows the plot between displacement and time for phase φ = 0.

t

T

x

FigurE 13.1

8. Velocity and Acceleration of SHM: As functions of time, the particle velocity [v(t)] and acceleration [a(t)] during SHM, respectively, are expressed as

v t A t

t A t t x t

( ) sin( );

( ) cos( ) ( ) ( ).

= − += − + ⇒ = −

ω ω φα ω ω φ α ω2 2

It is shown that both velocity and acceleration of a body executing SHM motion are periodic functions, having the velocity amplitude v Am = ω and acceleration ampli-tude a Am ,= ω2 respectively. Figure 13.2a shows the varia-tion of velocity with time in SHM with initial phase φ = 0.Figure 13.2b shows the variation of acceleration of particle in SHM with time having initial phase φ = 0.

t

T

+wA

−wA

(a)

(b)

t

T

+w 2A

−w 2A

FigurE 13.2 9. At any time, a particle executing SHM has potential

energy U = (1/2)kx2 and kinetic energy K = (1/2)mv2. If no friction exists, the mechanical energy of the system, that is, E = K + U always remains constant despite the fact that K and U change with time. Figure 13.3 shows the variation



of kinetic energy and potential energy of a harmonic oscillator with time where phase φ = 0.

EE

ner

gyPE + KE

T/2

PE

KE

T

FigurE 13.3

10. Some simple systems executing SHM:

a. Motion of a body suspended from a spring: A particle of mass m oscillating under the influence of a Hooke’s law restoring force given by

F = –kx,

where k is the force constant, exhibits simple harmonic motion with the following angular frequency (w) and period (T ):

ω

π

=

=

km

Tmk

;

.2

b. Simple pendulum: The motion of a simple pendulum moving back and forth through small angles is considered approximately to be simple harmonic and its period of oscillation is expressed as

Tlg

= 2π .

c. Compound pendulum: Compound pendulum is a rigid body (of any shape), which is capable of oscillating about a horizontal axis passing through it. For a compound pendulum executing SHM, the time period of oscillation is given by

TI

mg= 2π

.

11. For a SHM, when the damping force is given by

Fd = –bv,

where v is the velocity of the oscillator and b is a damp-ing constant, then the displacement of the oscillator is expressed as

x t A bt m( ) e cos( ’ ),/= +− 2 ω φ

where the angular frequency of the damped oscillator, w′, is expressed as

ω ’ .= −km

bm

2

24

When the damping constant is small, then ω ω’ ,≈ where w is the angular frequency of the undamped oscillator. The mechanical energy (E) of the damped oscillator is expressed as

E t kA bt m( ) e ./= −12

2

When an external force with angular frequency, wd, acts on an oscillating system with natural angular frequency, w, the system oscillates with angular frequency wd . The amplitude of oscillations is the highest when wd = w, which is a condition known as resonance of the oscillation.


Chapter 14Waves

TARGET CBSE

1. The patterns, which move without the real physical trans-fer or flow of matter as a whole, are called waves. For example, while we speak, the sound (known as sound waves) moves outward from us, without any flow of air from one part of the medium to another – the distur-bances produced in air are much less obvious and only our ears or a microphone can detect them.

2. There are mainly three types of waves exist: (a) mechanical waves, (b) electromagnetic waves, and (c) matter waves.

3. Mechanical waves obey Newton’s laws and they exist in material media.

4. All electromagnetic waves travel through vacuum at the same speed of light (c = 299,792,458 m/s).

5. Electromagnetic waves, unlike the mechanical waves, do not require any medium for their propagation.

6. The waves that are associated with moving electrons, protons, neutrons, other particles, and even with atoms and molecules are called matter waves as we commonly think of these as constituting matter.

7. When the constituents of a medium oscillate perpendicu-lar to the direction of wave propagation, it is called trans-verse waves.

8. When the constituents of a medium oscillate along the direction of wave propagation, it is called in longitudinal waves.

9. When a wave moves from one point of medium to another point, it is called traveling or progressive wave.

10. On the surface of water, the waves are of two types: (1) capillary waves and (2) gravity waves.

11. The displacement relation for a sinusoidal wave propa-gating in the positive x-direction is expressed as

y x t a kx t( , ) sin( ),= − +ω φ

where a is the amplitude of the wave, k is the angular wave number, w is the angular frequency, ( )kx t− +ω φ is the phase, and φ is the phase constant or phase angle.

12. The distance between two consecutive points of the same phase at a given time of a progressive wave is called wavelength, which is denoted by l. The wavelength in a stationary wave is twice the distance between the two consecutive nodes or antinodes.

13. Unit of wavelength is meter (m) and its dimension is [L].

14. The time taken by any element of the medium to move through one complete oscillation is called the period T

of oscillation of a wave, which is related to the angular frequency w by

T = 2πω

15. Frequency n of a wave is defined as 1/T, which is related to angular frequency by

ν ωπ

=2

.

16. Speed of a progressive wave:

vk T

= = =ω λ λν.

17. Speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with tension T and linear mass density m is expressed as

vT=µ

.

18. Sound wave is a longitudinal mechanical wave which travels through solids, liquids, or gases. The speed v of a sound wave in a fluid with bulk modulus (B) and density (r) is expressed as

v = Bρ

.

19. In a metallic bar, the speed of longitudinal waves is given by

vY=ρ

.

20. As B = g P, the speed of sound in gases is expressed as

vP= γ

ρ.

21. Principle of superposition of waves: In the same medium, when two or more waves traverse, the displace-ment of any element of the medium is the algebraic sum of the displacements due to each wave:

y f x tt

n

= −=

∑ 11

( ).υ

22. When two sinusoidal waves on the same string show inter-ference, adding or canceling according to the principle of superposition, and if the two are traveling in the same direction and have the same amplitude and frequency but differ in phase by a phase constant φ , the result is a single wave with the same frequency w :


Chapter 14 Waves64

y x t a kx t( , ) cos sin .=

− +

212

12

φ ω φ

In this case, if φ = 0 or an integral multiple of 2p, the waves are exactly in phase and the interference is constructive and if φ π= , they are exactly out of phase and the inter-ference is destructive.

23. A traveling wave, at a rigid boundary or a closed end, is reflected with a phase reversal, but the reflection at an open boundary takes place without any phase change. For an incident wave,

y x t a kx ti ( , ) sin( )= −ω

For the reflected wave at a rigid boundary is

y x t a kx tr ( , ) sin( )= − + ω

For reflection at an open boundary is

y x t a kx tr ( , ) sin( )= + ω

24. Standing waves are produced by the interference of two identical waves moving in opposite directions. For a string with fixed ends, the standing wave is expressed by

y x t a kx t( , ) [ sin ] cos= 2 ω

25. Standing waves are characterized by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The distance between the two consecutive nodes or antinodes is l /2. A stretched string of length L fixed at both the ends vibrates with frequencies given by

ν = =nvL

for n2

1 2 3, , , ,

in which, the set of frequencies are called the normal modes of oscillation of the system. The oscillation mode

with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscillation mode with n = 2, etc. For a pipe of length L with one end open and other end closed (such as air columns) vibrates with frequencies given by

ν = +

=nvL

for n12 2

0 1 2 3, , , , ,

in which the set of frequencies are the normal modes of oscillation of such a system. The lowest frequency given by v/4L is the fundamental mode or the first harmonic.

26. A string (of length L) fixed at both ends, or an air column closed at one end and open at the other end, vibrates with frequencies called its normal modes. Each of these fre-quencies is a resonant frequency of the system.

27. When two waves, having slightly different frequen-cies (n1 and n2) and comparable amplitudes, are super-posed, the outcome is called beats; the beat frequency is given by

ν ν νbeat .= −1 2

28. A change in the observed frequency of a wave, when the source and the observer move relative to the medium, is called Doppler effect. For sound, the observed frequency, n, is given in terms of the source frequency, n0 , by

ν ν= ++

0

0v vv vs

,

where v is the speed of sound through the medium, v0

is the velocity of observer relative to the medium, and vs is the source velocity relative to the medium. In using this formula, velocities in from the direction of observer to source should be treated as positive and those opposite to it should be taken to be negative.

TARGET COMPETITION

1. Longitudinal waves are those waves in which displacement or oscillations in medium are parallel to the direction of propagation of wave (e.g., sound waves).

2. Transverse waves are those waves where the displace-ments or oscillations are perpendicular to the direction of propagation of wave.

3. At any time t, displacement y of the particle from its equi-librium position as a function of the coordinate x of the particle is expressed as

y(x, t) = A sin(w t – kx),

where A is the amplitude of the wave, k is the wave number, w is angular frequency of the wave, and (wt – kx) is the phase.

4. Wavelength (l) and wave number (k) are related by the equation

k = 2πλ

.

5. Time period (T ) and frequency (n) of the wave are related to the angular frequency (w) by

ωπ2

1= =νT

.

6. Speed of a wave is expressed as

vk T

= = =ω λ λν.

7. Principle of superposition: When two or more waves traverse through the same medium, the displacement of any particle of the medium is the sum of the displacement that the individual waves would give it.

y = Σyi(x, t).

8. The interference of two identical waves moving in oppo-site directions produces standing waves.

9. For a string with fixed ends, the standing wave is expressed as

y(x, t) = [2A cos(kx)]sin (w t)



This equation does not represent traveling wave since it does not have characteristic form involving (wt – kx) or (wt + kx) in the argument of trigonometric function.

10. In standing waves, amplitude of waves is different at dif-ferent points, that is, at nodes, the amplitude is zero and at antinodes amplitude is maximum, which is equal to sum of amplitudes of constituting waves.

11. At intermediate points amplitude of wave varies between these two limits of maxima and minima.

12. Standing waves are produced by the interference of two identical waves moving in opposite directions. For a string with fixed ends, the standing wave is expressed by

y x t a kx t( , ) [ sin ] cos= 2 ω

13. Standing waves are characterized by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The distance between the two consecutive nodes or antinodes is l /2. A stretched string of length L fixed at both the ends vibrates with frequencies given by

ν = =nvL

for n2

1 2 3, , , ,

in which, the set of frequencies are called the normal modes of oscillation of the system. The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscil-lation mode with n = 2, etc. For a pipe of length L with one end open and other end closed (such as air columns) vibrates with frequencies given by

ν = +

=nvL

for n12 2

0 1 2 3, , , , ,

in which the set of frequencies are the normal modes of oscillation of such a system. The lowest frequency given by v/4L is the fundamental mode or the first harmonic.

14. A string (of length L) fixed at both ends, or an air column closed at one end and open at the other end, vibrates with frequencies called its normal modes. Each of these frequencies is a resonant frequency of the system.

15. When two waves, having slightly different frequencies (n1 and n2) and comparable amplitudes, are superposed, the outcome is called beats; the beat frequency is given by

ν ν νbeat .= −1 2

16. A change in the observed frequency of a wave, when the source and the observer move relative to the medium, is called Doppler effect. For sound, the observed frequency, n, is given in terms of the source frequency, n0 , by

ν ν= ++

0

0v vv vs

,

where n is the speed of sound through the medium, n0

is the velocity of observer relative to the medium, and ns is the source velocity relative to the medium. In using this formula, velocities in from the direction of observer to source should be treated as positive and those opposite to it should be taken to be negative.


Chapter Summaries For University Physics

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