Illustration – 1 : A particle of mass 2 kg moves under the action of a constant force = N. If its

displacement is 6 m. What is the work done by the force ?

Solution :The work done .

= . = - 12 Joule

Illustration – 2 : A load of mass m = 3000 kg is lifted by a rope with an acceleration a = 2 m/s 2. Find the work done during the first one and a half seconds from the beginning of motion.

Solution :

The height to which the body is lifted during the first 't' second is h = at2 tension in

the rope T = mg + ma

Work done = T.h = m(g +a) = 3000 (10 + 2)

= 81 KJ

Illustration – 3 : A train is moving with a constant speed "v". A box is pushed by a worker applying a force "F" on the box in the train slowly by distance "d" on the train for time "t". Find the work done by "F" from the train frame as well as from the ground frame.

Solution :As the box is seen from the train frame the displacement is only 'd' if the force direction is same as the direction of motion of the box. Then the work done = F.d = Fdcos00 = Fd

= Fdcos1800 = -Fd (if the displacement on the train is opposite to 'F')

As the box is seen from ground frame, the displacement of the box = vt + d (if the displacement is along the

direction of motion of the train ) = d - vt (if the displacement is opposite to

direction of motion of the train)then work done = F. (vt + d) = Fvt + Fd OR = F.(d-vt) = Fd - Fvt

Illustration – 4 : A block is (mass m) placed on the rough surface of a plank (mass m) of coefficient of friction "" which in turn is placed on a smooth surface. The block is given a velocity v0 with respect to the plank which comes to rest with respect to the plank. Find the a) The total work done by friction in the plank frame. b) The work done by friction on the smaller block in the plank frame.c) Find the final velocity of the plank

Solution :The acceleration of the plank = Friction force applied by the block on the plank / mass of the plank.

1

mm 0v

(a) Pseudo force acting on the block = g (back wards)Force of friction is mg ( acting backwards)From the plank frame time needed to stop the block is given by

O =

t =

Velocity of the plank during this time is

=

Displacement of the block = S =

Work done by friction on the block = =

(b) From the Plank frameWork done by friction on smaller block = -mgl

work done by friction from the Plank frame =

(c) Final velocity of the block

= Velocity of the plank =

Illustration – 5 :

The velocity of an 800 gm object changes from = 3 - 4 to = -6 + 2 m/s. What

is the change in K.E of the body?

Solution :Here m = 800gm = 0.8 kg

= = 5 =

change in K.E = x 0.8 =

Illustration – 6 : The coefficient of sliding friction between a 900 kg car and pavement is 0.8. If the car is moving at 25 m/s along level pavement, when it begins to skid to a stop, how far will it go before stopping?

Solution :Here m = 900kg = 0.8, v = 25 m/s S =?

K.E = work done against friction = F.s = N.s = mgs

s = = ~ 39 m

Illustration – 7 :

2

mg

pma

m

An object of mass 10kg falls from rest through a vertical distance of 20m and acquires a velocity of 10 m/s. How much work is done by the push of air on the object ? (g = 10 m/s2)

Solution :Let upward push of air be F

The resultant downward force = mg - F As work done = gain in K.E

(mg - F) x S =

(10 x 10 - F) x 20 = x 10 x (10)2 F = 75 N

Work done by push of air = 75 x 20 = 15 JouleThis work done is negative.

CONSERVATION OF MECHANICAL ENERGY : Change in potential energy U = - WC where WC is the work done by conservative

forces. From work energy theoremWnet = k

Where Wnet is the sum of work done by all the forces acting on the mass. If the system is subjected to only conservative forces then Wnet = WC = k

U = - k U + k = 0The above equation tells us that the total change in potential energy plus the total

change in kinetic energy is zero, if only conservative forces are acting on the system.(k+U) = 0 or E = 0 where E = k + U

When only conservative forces act, the change in total mechanical energy of a system is zero. i.e if only conservative forces perform work on and within a system, the total mechanical energy of the system is conserved.

kf + Uf - (ki + Ui) = 0 kf + Uf = ki + Ui

E = 0, integrating both sides E = constant.

3

Illustration – 8 : A projectile is fired from the top of a 40m. high cliff with an initial speed of 50 m/s at an unknown angle. Find its speed when it hits the ground.

Solution :Taking ground as the reference level we can conserve the mechanical energy between the points A and B

(K + U) = 0 Ki + Ui = Kf + Uf

mv2 + mgH = mv' 2 + 0

(50)2 + 40 x 10 = v' 2

(1250 + 400) x 2 = v' 2

v' 2 = 3300 v' ~ 58 m/s

Illustration – 9 : A car of mass 500 kg moving with a speed 36km/hr in a straight road unidirectionally doubles its speed in 1 minute. Find the average power delivered by the engine.

Solution :Its initial speed V1 = 10 m/s then V2 = 20 m/s

k = m

Power delivered by the engine

P =

=

= 1250 W.

4

'v

H

A

v

B

MOTION IN A VERTICAL CIRCLE :

A particle of mass 'm' is attached to a light and inextensible string. The other end of the sting is fixed at O and the particle moves in vertical circle of radius 'r' equal to the length of the string as shown in the fig. At the point P, net radial force on the particle is T-mg cos.

T - mg cos =

T = mg cos +

The particle will complete the circle if the string does not slack even at the highest point ( = ). Thus, tension in the string should be greater than or equal to zero (T > 0) at = for critical situation T = 0 and =

mg = =

= Now conserving energy between the lowest and the highest point

If the particle will complete the circle. At u = , velocity at highest point is v = and tension in the string is zero.

If u < , the tension in the string become zero before reaching the highest point and at that point the particle will leave the circular path. After leaving the circle the particle will follow a parabolic path.

Above conditions are applicable even if a particle moves inside a smooth spherical shell of radius R. The only difference is that the tension is replaced by the normal reaction N.

Illustration – 10 : A heavy particle hanging from a fixed point by a light inextensible string of length l is projected horizontally with speed . Find the speed of the particle and the inclination of the string to the vertical at the instant of the motion when the tension in the string is equal to the weight of the particle.

Solution :

Let T = mg at an angle as shown in figureh = l (1 - cos)Conserving mechanical energy between

A and B mu2 = mv2 + mgh

u2 = v2+ 2gh v2 = u2 - 2gh …. (i)

T - mg cos = T= mg cos +

mg = mg cos +

v2 = g l (1- cos) ……………. (ii)

From (i) and (ii) u2 - 2gl (1 - cos) = gl (1 - cos)

5

OT

Pcosmg

sinmg

A gu

cosmg

sinmg

B

h

cos = = cos-1

putting the value of cos in equation ………… (ii)

v2 = gl = v =

Illustration – 11: The P.E of a Conservative system is given as U = 10 + (x-2)2. Find the

equilibrium position and discuss type of equilibrium.

Solution: For Equilibrium F = 0

F =-

x = 2

and

it is Stable equilibrium position at x= 2 and P.E at that position is 20 units.

* * * * *

6

WORKED OUT OBJECTIVE PROBLEMS

EXAMPLE : 01A particle moves with a velocity 5 - 3 + 6 m/s under the influence of a constant force

= N. The instantaneous power applied to the particle isA) 200 J/S B) 40 J/S C) 140 J/S D) 170 J/S

Solution :P = . = (5 - +6 ) . (10 + 10 +20 ) = 50 - 30 + 120 = 140 J/S

EXAMPLE : 02

A 15 gm ball is shot from a spring gun whose spring has a force constant of 600 N/m. The spring is compressed by 5 cm. The greatest possible horizontal range of the ball for this compression is (g = 10 m/s2)

A) 6.0 m B) 12.0 m C) 10.0 m D) 8.0 m

Solution :

R max = = =

= .

[ Note : The actual value of 'u' will be less than the calculated value as some part of 1/2kx 2

is used up in doing work against gravity when the spring regains its length]

EXAMPLE : 03Force acting on a particle is (2 + 3 ) N. work done by this force is zero, when a particle is moved on the line 3y + kx = 5 Here value of k is

A) 3 B) 2 C) 1 D) 4Solution :

Force is parallel to the line y = 3/2 x + c

and the given line can be written as y =

as the work done is zero force is perpendicular to the displacement

= - 1

k = 2 EXAMPLE : 04

Power supplied to a particle of mass 2 kg varies with time as p = watt. Here

't' is in second. If velocity of particle at t = 0 is v = 0. The velocity of particle at time t = 2 second will be

A) 1 m/s B) 4 m/s C) 2 m/s D) 2 m/sSolution :

kf - ki = mv2 = t2 dt v2 =

m = 2 kg v = 2 m/s

EXAMPLE : 05

A particle of mass 'm' is projected with velocity 'u' at an angle with horizontal. During the period when the particle descends from highest point to the position

7

where its velocity vector makes an angle /2 with horizontal, work done by the gravity force is

A) 1/2 mu2 tan2 /2 B) 1/2 mu2 tan2 C) 1/2 mu2 cos2 tan2 /2 D) 1/2 mu2 cos2 /2 sin2

Solution :As horizontal component of velocity does not change v cos /2 = ucos

v =

Wgravity = K = mv2 - m (u cos)2

= mu2 cos2 tan2

EXAMPLE : 06

A body of mass 1 kg thrown upwards with a velocity of 10 m/s comes to rest (momentarily) after moving up 4 m. The work done by air drag in this process is (g = 10 m/s2)

A) 10 J B) - 10 J C) 40 J D) 50 J

Solution :From work energy theorem Wgr + Wair drag = k

- mgh + Wair drag = 0 - mu2

Wair drag = mgh - mu2 = (40 - 50) J = - 10 J

EXAMPLE : 07

The potential energy of particle of mass 'm' is given by U = kx2 for x < 0 and U

= 0 for x > 0. If total mechanical energy of the particle is E. Then its speed at x

= is

A) zero B) C) D)

Solution :

Potential energy of particle at x = is zero K.E = E

mv2 = E or v =

EXAMPLE : 08

A block is suspended by an ideal spring of force constant k. If the block is pulled down by applying a constant Force 'F' and if maximum displacement of block from its initial position of rest is then

A) < <

B) =

C) Work done by force F is equal to F

D) Increases in energy stored in spring is k2

Solution :

8

2/

cosu

V

u

If the mass of the hanging block be 'm' then elongation of spring is .

Due to the applied force the additional stretching is

F + mg = K

= K

= K 2 + mg = .

EXAMPLE : 09

A stone is projected at time t = 0 with a speed V0 and an angle with the horizontal in a uniform gravitational field. The rate of work done (P) by the gravitational force plotted against time (t) will be as

A) B) C) D)

Solution :Rate of work done is the power associated with the force. It means rate of work done by the gravitational force is the power associated with the gravitational force. Gravitational force acting on the block is equal to its weight mg which acts vertically downwards.Velocity of the particle (at time t) has two components,(i) a horizontal component v cos and(ii) a vertically upward component (v sin - gt)Hence, the power associated with her weight mg will be equal to p = m . = -mg (v sin - gt)This shows that the curve between power & time will be straight line having positive slope but negative intercept on Y-axis. Hence (D) is correct.

9

P

O t

P

O t

P

O t tO

P

SINGLE ANSWER TYPELEVEL – I 1. Two springs A and B(KA = 2KB) are stretched by applying forces of equal magnitudes at

the four ends. If the energy stored in A is E, that in B isa) E/2 b) 2E c) E d) E/4

2. Two equal masses are attached to the two ends of a spring constant K. The masses are pulled out symmetrically to stretch the spring by a length x over its natural length. The work done by the spring on each mass isa) ½ Kx2 b) -1/2 Kx2 c) ¼ Kx2 d) -1/4 Kx2

3. The negative of the work done by the conservative internal forces on a system equals the change in a) total energy b) kinetic energy c) potential energy d) none of these

4. The work done by the external forces on a system equals the change ina) total energy b) kinetic energy c) potential energy d) none of these

5. The work done by all the forces (external and internal) on a system equals the change ina) total energy b) kinetic energy c) potential energy d) none of these

6. ________ of a two particle system depends only on the separation between the two particles. The most appropriate choice for the blank space in the above sentence isa) kinetic energy b) total mechanical energy c) potential energyd) total energy

7. A small block of mass ‘m’ is kept on a rough inclined surface of inclination fixed in an elevator. The elevator goes up with a uniform velocity ‘v’ and the block does not slide on the wedge. The work done by the force of friction on the block in time ‘t’ will bea) zero b) mgvt cos2 c) mgvt sin2 d) mgvt sin2

8. A block of mass ‘m’ slides down a smooth vertical circular track. During the motion, the block is ina) vertical equilibrium b) horizontal equilibrium c) radial equilibriumd) none of these

9. A particle is rotated in a vertical circle by connecting it to a string of length ‘ l’ and keeping the other end of the string fixed. The minimum speed of the particle when the string is horizontal for which the particle will complete the circle is a) b) c) d)

10.Consider two observers moving with respect to each other at a speed v along a straight line. They observe a block of mass m moving a distance ‘l’ on a rough surface. The following quantities will be same as observed by the two observersa) kinetic energy of the block at time t b) work done by frictionc) total work done on the block d) acceleration of the block

11.A particle of mass ‘m’ is attached to a light string of length ‘ l’, the other end of which is fixed. Initially the string is kept horizontal and the particle is given an upward velocity v. The particle is just able to complete a circlea) the string becomes slack when the particle reaches its highest pointb) the velocity of the particle becomes zero at the highest pointc) the kinetic energy of the ball in initial position was ½ mv2 = mgld) the particle again passes through the initial position

12.The kinetic energy of a particle continuously increases with timea) the resultant force on the particle must be parallel to the velocity at all instantsb) the resultant force on the particle must be at an angle less than 900 all the timec) its height above the ground level must continuously decreased) the magnitude of its linear momentum is increasing continuously

13.One end of a light spring of spring constant k is fixed to a wall and the other end is tied to a block placed on a smooth horizontal surface. In a displacement, the work done by the spring is ½ kx2. The possible cases area) the spring was initially compressed by a distance x and was finally in its natural length b) it was initially stretched by a distance x and finally was in its natural length c) it was initially in its natural length and finally in a compressed position

10

d) it was initially in its natural length and finally in a stretched position14. A block of mass ‘M’ is hanging over a smooth and light pulley through a light string.

The other end of the string is pulled by a constant force F. The kinetic energy of the block

increases by 20J in 1s

a) the tension in the string is Mg b) the tension in the string is F c) the work done by the tension on the block is 20J in the above is 1s d) the work done by the force of gravity is –20J in the above 1s

15. A particle of mass 0.25kg moves under the influence of a force F = (2x-1). If the velocity of the particle at x = 0 is 4m/s. its velocity at x = 2m will beA) 4 m/s B) 2 m/s C) 8m/s D) 6m/s

16. Work done to accelerate a car from 10 to 20m/s compared with that required to accelerate it from 0 to 10m/s isA) twice B) three times C) four times D) same

17. Two springs have their force constant as K1 and K2 (K1 > K2). When they are stretched by the same force :A) no work is done in case of both the springs B) equal work is done in case of both the springsC) more work is done in case of second spring D) more work is done in case of first spring

18. The kinetic energy K of a particle moving in a straight line depends upon the distance s as K = as2 where a is a constant. The force acting on the particle isA) 2as B) 2mas C) 2a D)

19. A particle moves in a straight line with a retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional toA) x B) x2 C) ln x D) ex

20. A particle falls from rest under gravity. Its potential energy (PE) with respect to the ground and its kinetic energy (KE) are plotted against time (t). Choose the correct graph.

A) B) C) D) 21. Choose the wrong option

A) If conservative forces are doing negative work then potential energy will increase and kinetic energy will decrease.B) If kinetic energy is constant it means work done by conservative forces is zero.C) for change in potential energy only conservative forces are responsible, but for change in kinetic energy other than conservative forces are responsibleD) all of the above are wrong

22. Instantaneous power of a constant force acting on a particle moving in a straight line under the action of this force :A) is constant B) increases linearly with timeC) decreases linearly with time D) either increases or decreases linearly with time.

23. Suppose y represents the work done and x the power, then dimensions of will be

:A) B) C) D)

24. Choose the correct statement Work done by a variable forceA) Is defined as B) Is independent of path

11

C) Is always dependent on the initial and final positions D) None of these25. Identify the correct statement for a non-conservative force

A) A force which is not conservative is called a non-conservative forceB) The work done by this force depends on the path followedC) The word done by this force along a closed path is zeroD) The work done by this force is always negative

26. The figure shows a plot of potential energy function, u(x) = kx2 where x is the displacement and k is a constant. Identify the correct conservative force function F(x)

27. A plot of velocity versus time is shown in figure. A single force acts on the body. Find correct statement A) In moving from C to D, work done by the force on the body is positiveB) In moving from B to C, work done by the force on the body is positiveC) In moving from A to B, the body does work on the system and is negativeD) In moving from O to A, work done by the body and is negative

28. The force acting on a body moving along x-axis varies with the position of the particle as shown in the figure. The body is in stable equilibrium at A) x = x1 B) x = x2

C) both x1 and x2 D) neither x1 and x2

29. Displacement time graph of a particle moving in a straight line is as shown in figure. Select the correct alternative(s). A) Work done by all the forces in region OA and BC is positiveB) Work done by all the forces in region AB is zeroC) Work done by all the forces in region BC is negativeD) Work done by all the forces in region OA is negative

KEY1 2 3 4 5 6 7 8 9 10 11 12 13 14 15B D C A B C C D C D AD BD AB B A16 17 18 19 20 21 22 23 24 25 26 27 28 29B C A B B D B A C B B A B B

LEVEL - II1. A particle of mass m is moving in a circular path of radius r under the influence of

centripetal force F – C/r2. The total energy of the particle is

a) b) c) C x 2r d) Zero

Sol: Fcentipetal F = ; v = ; E1 = EK + v = C/2r – C/r

= -C/2r2. Water from a stream is falling on the blades of a turbine at the rate of 100kg/sec.

If the height of the stream is 100m then the power delivered to the turbine is

12

a) 100 kw b) 100 w c) 10 kw d) 1 kwSol: P = w/1 = (m/g) gh = 100 x 10 x 100 = 105w

3. A body is being moved along a straight line by a machine delivering a constant power. The distance covered by the body in time t is proportional to a) b) t3/2 c) t3/4 d) t2

Sol: P = Fv = constant or ma . at = constant or a2t = constant S = 1/2at2 or S at2 But a 1/ S t2/ or S t3/2

4. A ball is dropped from a height of 10m. If 40% of its energy is lost un collision with the earth then after collision the ball will rebound to a height of a) 10m b) 8m c) 4m d) 6m

Sol:

5. A particle moves under the influence of a force F = CX from X = 0 to X = X1. The work done in this process will be

a) b) c) d) 0

Sol: W =

6. A uniform chain of mass M and length L lies on a horizontal table such that one third of its length hangs from the edge of the table. The work done is pulling the hanging part on the table will be

a) b) MgL c) d)

Sol: W = M/3 . g . 1/67. A body of mass 2kg moves under the influence of a force. Its position x changes

with time according to the relation x = t3/3 where x is in meter and t in seconds. The work done by this force in first two seconds will bea) 1600 Joule b) 160 Joule c) 16 Joule d) 1.6 JouleSol: W = ½ mv2

2 – ½ mv12

8. A man and a child are holding a uniform rod of length L in the horizontal direction in such a way that one fourth weight is supported by the child. If the child is at one end of the rod then the distance of man from another end will be a) 3L/4 b) L/4 c) L/3 d) 2L/3

Sol:

9. An electric motor produces a tension of 4500N in a load lifting cable and rolls it at the rate of 2m/s. The power of the motor is a) 9 kw b) 15 kw c) 225 kw d) 9 x 103 HPSol: P = Fv = 4500 x 2 = 9 kw

10. A body of mass m is accelerated to velocity v in time et1. The work done by the force as a function of time t will be

a) b) c) d)

Sol: Acceleration produced in a body a = ; W = ma2t2 =

11. A motor of 100 HP is moving with a constant velocity of 72 km/hour. The forward force exerted by the engine of the car is

13

a) 3.73 x 103 N b) 3.72 x 102 N c) 3.73 x 101 N d) None of the aboveSol: F = P/v

12. The kinetic energy of a man is half the kinetic energy of a boy of half of his mass. If the man increases his speed by 1m/s, then his kinetic energy becomes equal to that of the boy. The ratio of the velocity of the boy and that the man is a) 2/1 b) 1/2 c) 3/4 d) 4/3

Sol: According to question

13. A bomb of mass 9 kg explodes into 2 pieces of 3kg and 6kg. The velocity of 3 kg piece is 16 m/s. The kinetic energy of 6kg piece isa) 768 Joule b) 786 Joule c) 192 Joule d) 687 Joule

Sol: m1v1 = m2v2;

14. The increase in the potential energy of a body of mass m, when it is carried from the surface of earth upto a height equal to the radius of earth Re, will bea) mgRe b) mgRe/2 c) mgRe/4 d) 2mgRe

Sol:

15. A person of mass 60kg carries a 15 kg body on the top of a building 10m high in 3 minutes. His efficiency is a) 40% b) 30% c) 20% d) 10%

Sol: M =

16. A force = (3x2 + 2x – 7)N acts on a 2 kg body as a result of which the body gets displaced form x = 0 to x = 5m. The work done by the force will bea) 35 Joule b) 70 Joule c) 115 Joule d) 270 Joule

Sol: W =

17. A 50 gm bullet moving with a velocity of 10 m/s gets embedded into a 950 gm stationary body. The loss in kinetic energy of the system will bea) 5% b) 50% c) 100% d) 95%

Sol:

18. A crane lifts 300 kg weight from earth’s surface upto a height of 2m in 3 seconds. The average power generated by it will bea) 1960 watt b) 2205 watt c) 4410 watt d) 0 wattSol: P = w/t = mgh/t

19. A block of mass 16kg is moving on a frictionless horizontal surface with velocity 4m/s and comes to rest after pressing a spring. If the force constant of the spring is 100 N/m then the compression in the spring will bea) 3.2 m b) 1.6 m c) 0.6 m d) 6.1 mSol: ½ mv2 = ½ kx2

14

20. The relation between time and displacement of a particle moving under the influence of a force F is t = +3 where x is in meter and t in second. The displacement of the particle when its velocity is zero will bea) 1 m b) 0 m c) 3 m d) 2 mSol: t = + 3 or x = (t – 3)2; v = dx/dt

21. A 0 kg satellite completes one revolution around the earth at a height of 100 km in 108 minutes. The work done by the gravitational force of earth will be

a) 108 x 100 x 10 Joule b) Joule c) 0 Joule d)

Joule

Sol: W = Fd cos = Fd cos 900 = 022. A particle moves in a potential region given by u = 8x2 – 4x

+ 400 Joule. Its state of equilibrium will bea) x = 25 m b) x = 0.25 m c) x = 0.025 m d) x = 2.5 mSol: F = - du/dx

23. Two men with weights in the ratio 5 : 3 run up a stair case in time in the ratio 11 : 9. The ratio of power of first to that of second is a) 15/11 b) 11/15 c) 11/9 d) 9/11

Sol: P =

24. A moving particle of mass m collides head on with another stationary particle of mass 2m. What fraction of its initial kinetic energy will m lose after the collision?a) 9/8 b) 8/9 c) 19/18 d) 18/19

Sol: mu + 2m x 0 = (m + 2m)v;

25. The potential energy function of a diatomic molecule is

given as u(r) = , where a and b are positive constants and r is inter atomic

distance. The equilibrium between two atoms is

a) b) c) d)

Sol:

26. A pump pulls 1000 kg water per minute from a 15 m deep well and provides 4 m/s velocity to it. The power of pump is (g = 10 m/s2)a) 2.6 kw b) 2.6 w c) 0.6 w d) 0.6 kw

Sol:

27. A body weighing 80N is moved up a slope of angle 600 with the horizontal through a displacement of 1m. The energy loss due to friction is 20%. The energy gained by the body will bea) J b) 64 J c) J d) 80 JSol: W = mg sin d

28. For the path PQR in a conservator force field (figure) amounts work done in carrying a body from P to Q and from Q to R are 5 Joule and 2 Joule respectively. The work done in carrying the body from P to R will be

15

a) 7 Jouleb) 3 Joulec) Jouled) ZeroSol: WPR = WPQ + WQR

29. Two particles each of mass m and traveling with velocities u1 and u2 collide perfectly inelastically. The loss of energy will bea) ½ m(u1 – u2)2 b) ¼ m(u1 – u2)2 c) m(u1 – u2)2 d) 2m(u1 – u2)2

Sol: E =

30. Two protons are situated at a distance of 100 fermi from each other. The potential energy of this system will be in eva) 44 b) 1.44 x 103 c) 1.44 x 102 d) 1.44 x 104

Sol: U = kq2/r31. In order to reduce the kinetic energy of a body to half its

initial value, its speed will have to be changed by the following factor, of its initial speeda) 1/ times b) times c) 1/2 times d) 2 timesSol: E = ½ mv2; v =

32. A body of mass M and moving with velocity u makes a head on elastic collision with another stationary body of m. If A = m/M, then the ratio (f) of the loss of energy of M to its initial energy will be

a) f = A(A + 1)2 b) f = c) f = d) f =

Sol: f =

33. Two masses m1 = 2kg and m2 = 5kg are moving on a frictionless surface with velocities 10 m/s and 3 m/s respectively. m2 is ahead of m1. An ideal spring of spring constant k = 1120 N/m is attached on the backside of m2. The maximum compression of the spring will be, if on collision the two bodies stick together.a) 0.51 m b) 0.062 mc) 0.25 m d) 0.72 m

Sol: E =

34. A body at rest explodes all of a sudden in three equal parts. The moments of two parts are Pi and 2Pj and their kinetic energies are k1 and k2. If and k3 are the momentum and kinetic energy respectively of the third part then the ratio k2/k3 will bea) 2/5 b) 3/5 c) 4/5 d) 1/5Sol: Conceptual

35. A block falls down from a table 0.5m high. It falls on an ideal vertical spring of constant 4 x 102 N/m. Initially the spring is 25 cm long and its length becomes 10 cm after compression. The mass of the block is (g = 10m/s2)a) 0.5 kg b) 2 kg c) 1.2 kg d) 0.9 kgSol: mgh = ½ kx2

36. The mass of a bucket full of water is 15 kg. It is being pulled up from a 15m deep well. Due to a hole in the bucket 6 kg water flows out of the bucket. The work done in drawing the bucket out of the well will be

16

a) 900 joule b) 1500 joule c) 1800 joule d) 2100 joule

Sol: W = mgh =

37. A spring of force constant k is first stretched by a lens x and then again by a further length x. The work done in the first case is w1 and in the second case w2, thena) w2 = w1 b) w2 = 2w1 c) w2 = 3w1 d) w2 = 4w1

Sol: w1 = ½ kx2, w3 = ½ k(2x2)38. A 2k body is projected, at an angle of 300 with the

horizontal, with a velocity of 10m/s. The kinetic energy of the body after 1 second will bea) 10 joule b) 50 joule c) 100 joule d) 200 jouleSol: v =

39. A 10 kg block is pulled in the vertical plane along a frictionless surface in the form of an arc of a circle of radius 10m. The applied force is of 200N as shown in the figure. If the block started from rest to A, the velocity at B would bea) 1.732 m/s b) 17.32 m/sc) 173.2 m/s d) none of theseSol: ½ mx2 = 200 cos 300 x

40. A block of mass m is pushed towards a movable wedge of mass m and height h with a velocity u. All surfaces are smooth. When the block collides with the wedge, the velocity of centre of mass of block wedge system will be

a) u b)

c) u(1 + ) d) 0Sol: mu = (m + m)v cm

41. In the above problem, the minimum value of u for which the block will reach the top of the wedge, will be

a) b) c) d)

Sol: ½ mu2 = mgh + ½ (m + m)V2cm42. A liquid in a U tube is changed from position (a) to position

(b) with the help of a pump. The density of liquid is d and area of cross section of the tube is a. The work done in pumping the liquid will bea) dghab) dgh2ac) 2gdh2ad) 4dgh2aSol: W = 2ahdgh – 2(ahdg h/2) = dgh2a

43. The human heart discharges 75cc of block through the arteries at each beat against an average pressure of 10cm of mercury. The pulse frequency of the heart is 72 per minute. The rate of working of heart isa) 2.35 w b) 3.29 w c) 1.19 w d) 9.11 w

Sol: P = hdg

44. A block of mass 1kg is pulled up on an incline of angle 300

with the horizontal. The block moves with an acceleration of 1 m/s2. The power delivered by the pulling force at t = 4s will bea) 12 w b) 36 w c) 24 w d) 48 w

17

Sol: F – mg sin = ma or F = mg sin + ma45. A particle of mass m is moving in a circular path of constant

radius r. The centripetal acceleration of the particle (ac) is varying with time t according to following relation ac = k2n2 where k is a constant. The power delivered to the particle by the forces acting on it will be a) mk2 r2 t2 b) m2k2 r2t2 c) m2k2 rt d) mk2r2tSol: ac = v2/r = k2n2; w = ½ mv 2/2 – ½ mv1

2 = ½ m k2r2t2 = 0; P = dw/dt46. A block of mass 2kg is released from A on a track that is a

on quadrant of a circle of radius 1m. It slides down the track and reaches B with a speed of 4m/s and finally stops at C at a distance of 3m from B. The work done against the force of friction is a) 2 joule b) 5 joulec) 10 joule d) 20 joule

Sol: W =

47. A man pulls a bucket full of water from a h metre deep well. If the mass of the rope is m and mass of bucket full of water is M, then the work done by the man is

a) b) c) d) (M + m) gh

Sol:

48. A particle has shifted along some trajectory in the x – y plane from point to another point . During that time, the

particle experiences the action of two forces . The work done by the forces on the particle will bea) 5 joule b) -5 joule c) 10 joule d) -10 jouleSol:

49. A 2kg body is dropped from height of 1m on to a spring of spring constant 800 kg/m as shown in the figure. A frictional force equivalent to 0.4 kg wt acts on the body. The speed of the body just before striking the spring will be a) 1 m/s b) 2 m/sc) 3 m/s d) 4 m/sSol: mgh = ½ mv2 + Ffr h

50. A shell is fired from a cannon with a velocity v and at an angle from the horizontal to hit a target at a horizontal distance R. It splits in two equal parts at the highest point of its path. One part refracts its path and reaches back upto the cannon. The velocity of the second part just after the explosion will bea) 3/2 v cos b) 2 v cos c) 3 v cos d) /2 v cos Sol: mv cos = m/2 v cos + m/2 v

51. A block of mass 10 kg moving on a smooth surface with a speed of 30 m/s bursts into two equal parts. Both parts continue to move in the seme direction. If one of the parts moves at 40 m/s, the energy produce in the process isa) 200 J b) 500 J c) 700 J d) JSol: mv = m1 v1 + m2 v2; E = ½ m1v1

2 + ½ m2 v22 – ½ mv2

18

52. Two identical 5 kg blocks are moving with same speed of 2 m/s towards each other along a frictionless horizontal surface. The two blocks collide, stick together and come to rest. The work done by the external forces is a) 0 b) 10 J c) 20 J d) none of these Sol: As Fext = 0; Wext =

53. In the above problem, the work done by the inertial forces is a) 0 b) 10 J c) 20 J d) none of these Sol: Wint = ½ mv2 + ½ mv2

54. The force-displacement curve for a body moving on a smooth surface under the influence of foce F acting along the direction of displacement s has been shown in fig. If the initial kinetic energy of the body is 2.5J. its kinetic energy at s = 6m is A) 7J B) 4.5J C) 2.25J D) 9J

55. A bullet, moving with a speed of 150m/s, strikes a wooden plank. After passing through the plank its speed becomes 125m/s. Another bullet of the same mass and size strikes the plank with a speed of 90m/s. It speed after passing through the plank would beA) 25m/s B) 35m/s C) 50m/s D) 70m/s

56. A man of mass 60kg climbs a staircase inclined at 450 and having 10steps. Each step is 20cm high. He takes 2 seconds for the first five steps and 3 seconds for the remaining five steps. The average power of the man isA) 245W B) 245 W C) 235 W D) 235W

57. The potential energy of a particle moving in x-y plane is given by U = x2 + 2y. The force acting on the particle at (2, 1) isA) 6N B) N C) N D) 0

58. Water is flowing in a river at 20m/s. The river is 50m wide and has an average depth of 5m. The power available from the current in the river isA) 0.5MW B) 1.0MW C) 1.5MW D) 2.0MW

59. A 5kg brick of dimensions 20cm x 10cm x 8cm is lying on the largest base. It is now made to stand with length vertical. If g = 10m/s2, then the amount of work done isA) 3J B) 5J C) 7J D) 9J

60. The displacement x of a particle moving in one dimension, under the action of a constant force is related to time t by the equation t = +3, where x is in metres and t in seconds. The work done by the force in first 6 seconds isA) 9J B) 6J C) 0J D) 3J

61. A body of mass m was slowly pulled up the hill by a force F which at each point was directed along the tangent of the trajectory. All surfaces are smooth. Find the work performed by this force A) mg B) -mg C) mgh D) zero

62. A rope ladder with a length l carrying a man of mass m at its end, is attached to the basket of a balloon of mass M. The entire system is in equilibrium in air. As the man climbs up the ladder into the balloon, the balloon descends by height h. Then the potential energy of manA) increases by mg l B) increases by mg (l -h)C) increases by mgh D) increases by mg (2 l -h)

63. Two springs s1 and s2 have negligible masses and the spring constant of s1 is one-third that of s2. When a block is hung from the springs as shown, the springs came to the equilibrium again. The ratio of work done is stretching s1

to s2 isA) 1/9

19

B) 1/3 C) 1D) 3

64. A light spring of length l and spring constant 'k' it is placed vertically. A small ball of mass m falls from a height h as measured from the bottom of the spring. The ball attaining to maximum velocity when the height of the ball from the bottom of the spring isA) mg/k B) l-mg/k C) l + mg/k D) l - k/mg

65. A block of mass 1kg is permanently attached with a spring of spring constant k = 100N/m. The spring is compressed 0.20m and placed on a horizontal smooth surface. When the block is released, it moves to a point 0.4m beyond the point when the spring is at its natural length. The work done by the spring in changing from compressed state to the stretched state isA) 10J B) -6J C) -8J D) 18J

66. A chain of length l and mass m lies on the surface of a smooth sphere of radius R with one end tied on the top of the sphere. If = R/2, then the potential energy of the chain with reference level at the centre of sphere is give byA) m R g B) 2m R g C) 2/ m R g D) 1/ m R g

67. If the force acting on a particle is given by = 2i + xyj + xz2k, how much work is done when the particle moves parallel to Z-axis from the point (2, 3, 1) to (2, 3, 4) ?A) 42J B) 48J C) 84J D) 36J

68. A uniform chain of length ' ' and mass m is placed on a smooth table with one-fourth of its length hanging over the edge. The work that has to be done to pull the whole chain back onto the table is

A) mgl B) mgl C) mgl D) mgl

69. A spring, which is initially in its unstretched condition, is first stretched by a length x and then again by a further length x. The work done in the first case is W 1 and in the second case is W2

A) W2 = W1 B) W2 = 2W1 C) W2 = 3W1 D) W2 = 4W1

70. A particle of mass m is fixed to one end of a light rigid rod of length ' ' and rotated in a vertical circular path about its other end. The minimum speed of the particle at its highest point must be1) zero B) C) D)

71. A force F acting on a body depends on its displacement x as F xn. The power

delivered by F will be independent of x if n isA) 1/3 B) -1/3 C) 1/2 D) -1/2

72. A particle is moving in a conservative force field from point A to B. UA and UB are the potential energies of the particle at points A and B and Wc is the work done in the process of taking the particle from A to B.A) Wc = UB - UA B) Wc = UA - UB C) UA > UB D) UB > UA

73. A force is given by Mv2/r when the mass moves with speed v in a circle of radius r. The work done by this force in moving the body over upper half circle along the circumference isA) zero B) C) Mv2 D) Mv2 /2

74. A moving railway compartment has a spring of constant 'k' fixed to its front wall. A boy in the compartment stretches this spring by distance x and in the mean time the compartment moves by a distance s. The work done by boy w.r.t earth is

A) B) (kx) (s+x) C) D)

75. Force acting on a block moving along x-axis is given by :

F =

The block is displaced from x=-2m to x=+4m, the work done will be

20

A) positive B) negativeC) zero D) may be positive or negative

75. The system is released from rest with both the springs in unstretched positions. Mass of each block is 1 kg and force constant of each springs is 10 N/m. Extension of horizontal spring in equilibrium is:A) 0.2m B) 0.4m C) 0.6m D) 0.8m

77. In a projectile motion, if we plot a graph between power of the force acting on the projectile and time then it would be like :

A) B) C) D) 78. A golfer rolls a small ball with speed u along the floor from point A. If x = 3R,

determine the requiredspeed u so that the ball returns to A after rolling on the circular surface in the vertical plane from B to C and becoming a projectile at C. (Neglect friction)

A) B)

C) D) none of these

79. A wind-powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical. For wind speed v, the electrical power output will be proportional toA) v B) 2 C) 3 D) 4

KEY54 55 56 57 58 59 60 61 62 63 64 65 66 67 68A B D B B A C C B D B B C A D69 70 71 72 73 74 75 76 77 78 79C A B B A A B B B B C

LEVEL – III1. A block m is pulled by applying a force F as shown in fig. If the block

has moved up through a distance 'h', the work done by the force F is A) 0 2) Fh

C) 2Fh D) Fh

2. A body of mass m, having momentum p is moving on a rough horizontal surface. If it is stopped in a distance x, the coefficient of friction between the body and the surface is given byA) = p/(2mg x) B) = p2 / (2mg s) C) = p2 / (2g m2s)D) = p2 (2g m2s2)

3. A body of mass m moves from rest, along a straight line, by an engine delivering constant power P. the velocity of the body after time t will be

A) B) C) D)

4. The spring shown in fig has a force constant k and the mass of block is m. Initially, the spring is unstretched when the block is released. The maximum elongation of the spring on the releasing the mass will be

A) B) C) 2 D) 4

21

5. A skier starts from rest at point A and slides down the hill, without turning or braking. The friction coefficient is . When he stops at point B, his horizontal displacement in S. The height difference h between points A and B isA) h = S/ B) h = SC) h = S2 D) h = S/2

6. A small block of mass m is kept on a rough inclined surface of inclination fixed in an elevator. The elevator goes up with a uniform velocity v and the block does not slide on the wedge. The work done by the force of friction on the block in time t will beA) zero B) mg vt cos2 C) mg vt sin2 D) mg vt sin2

7. A block of mass m starts at rest at height h on a frictionless inclined plane. The block slides down the plane travels a total distance d across a rough horizontal surface with coefficient of kinetic friction k and compresses a spring with force constant k, a distance x before momentarily coming to rest. The spring then extends and the block travels back across the rough surface, sliding up the plane. The maximum height h' that the block reaches on its return is

A) h' = h - 2d B) h' = h - 2d - kx2

C) h' = h - 2d + kx2 D) h' = h - 2d - kx2

8. A chain of length 3 and mass m lies at the top of smooth prism such that its length is one side and 2 is on the other side of the vertex. The angle of prism is 1200 and the prism is not free to move. If the chain is released. What will be its velocity when the right end of the chain is just crossing the top-most point?

A) B) C) D)

9. If a constant power P is applied in a vehicle, then its acceleration increases with time according to the relation

A) a = B) a = C) a = D) a =

10. A body of mass m slides downward along a plane inclined at an angle . The coefficient of friction is . The rate at which kinetic energy plus gravitational potential energy dissipates expressed as a function of time isA) mtg2 cos B) mtg2 cos (sin - cos )C) mtg2 sin D) mtg2 sin (sin - cos )

11. The potential energy for a force field is given by U(x, y) = sin (x + y). The force acting on the particle of mass m at (0, /4) isA) 1 B) C) 1/ D) 0

12. A uniform rope of length ' ' and mass m hangs over a horizontal table with two third part on the table. The coefficient of friction between the table and the chain is . The work done by the friction during the period the chain slips completely off the table isA) 2/9 mgl B) 2/3 mgl C) 1/3 mgl D) 1/9 mgl

13. A particle is moving in a force field given by potential U = - (x + y + z) from the point (1, 1, 1) to (2, 3, 4). The work done in the process isA) 3 B) 1.5 C) 6 D) 12

22

14. A compressed spring of spring constant k releases a ball of mass m. If the height of spring is h and the spring is compressed through a distance x, the horizontal distance covered by ball to reach ground is

A) x B)

C) x D)

15. A block of mass m = 2kg is moving with velocity vo towards a massless unstretched spring of force constant K = 10 N/m. Coefficient of friction between the block and the ground is = 1/5. Find maximum value of vo so that after pressing the spring the block does not return back but stops there permanently.A) 6 m/s B) 12m/s C) 8m/s D) 10m/s

16. Potential energy of a particle moving along x-axis under the action of only conservative forces is given as : U = 10 + 4 sin(4x). Here U is in Joule and x in meters. Total mechanical energy of the particle is 16J. Choose the correct option.A) At x = 1.25m, particle is at equilibrium position. C) both A and B are correctB) Maximum kinetic energy of the particle is 20J D) both A and B are wrong.

17. A system shown in figure is released from rest. Pulley and spring is massless and friction is absent everywhere. The speed of 5 kg block when 2 kg block leaves the contact with ground is (Take force constant of spring k = 40 N/m and g = 10 m/s2) A) m/s` B) 2 m/sC) 2m/s D) 4 m/s

18. Two blocks of masses m1 = 1 kg and m2 = 2 kg are connected by a non-deformed light spring. They are lying on a rough horizontal surface. The coefficient of friction between the blocks and the surface is 0.4 what minimum constant force F has to be applied in horizontal direction to the block of mass m1 in order to shift the other block? (g = 10 m/s2) A) 8 N B) 15 N C) 10 N D) 25 N

19. A block of mass m is attached with a massless spring of force constant k. The block is placed over a rough inclined surface for which the coefficient of friction is = ¾. The minimum value of M required to move the block up the plane is (Neglect mass of string and pulley and friction in pulley).

A) 3/5m B) 4/5m C) 6/5m D) 3/2m20. A particle of mass m is moving in a circular path of constant radius r such that its

centripetal acceleration ac is varying with time t as, ac = k2 r t2 where k is a constant. What is the power delivered to the particle by the forces acting on it?

A) 2 pmk2r2t B) mk2r2t C) D) zero

21. A particle, which is constrained to move along the x- axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin asF(x) = -kx + ax2. Here k and a are positive constant. For x 0, the function form of the potential energy (x) of the particle is

KEY

23

A) B) c) D)

1 2 3 4 5 6 7 8 9 10 11C C A C B C A B D B A12 13 14 15 16 17 18 19 20 21A C C D A B A A B

MULTIPLE ANSWER TYPE QUESTIONS

1. The potential energy U for a force field is such that U = - kxy, where k is a constantA) B) C) The force is a conservative force D) The force is a non-conservative force

2. A sledge moving over a smooth horizontal surface of ice at a velocity v0 drives out on a horizontal road and comes to a halt as shown. The sledge has a length l, mass m and friction between runners and road is A) No work is done by the friction to switch the sledge from ice to the road

B) A work of mgl is done against friction while sledge switches completely on to

road

C) The distance covered by the sledge on the road is

D) Total distance moved by the sledge before stopping is

3. A strip of wood of mass M and length l is placed on a smooth horizontal surface. An insect of mass m starts at one end of the strip and walks to the other end in time t, moving with a constant speed

A) The speed of the insect as seen from the ground is <

B) The speed of the strip as seen from the ground is

C) The speed of the strip as seen from the ground is

D) The total kinetic energy of the system is (m + M)

4. Two blocks A and B each of mass m are connected by a light spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A to B and collides with A. ThenA) the kinetic energy of the A-B system at maximum compression of the spring is zeroB) the kinetic energy of the A-B system at maximum compression of the spring is

C) the maximum compression of the spring is

24

D) the maximum compression of the spring is

5. Suppose a car is modeled as a cylinder moving with a speed v. If A is the area of cross section of the car and is the density of air then

A) Power loss due to air resistance is Av3 B) power loss due to air resistance is

Av3

C) drag force is Av2 D) drag force is Av2

6. A heavy mass M resting on the ground is connected to a lighter mass m through a light inextensible string passing over a light, frictionless pulley as shown in the figure. The string connected to mass M is loose. Let lighter mass m be allowed to fall freely through a height h such that the string becomes taut. If t is the time from this instant onwards when the heavier mass again makes contact with the ground and E is the change in kinetic then

A) t = B) t = C) E = - D) E

= -

7. The kinetic energy of a body moving along a straight line varies directly with time t. If the velocity of the body is v at time t, then the force F acting on the body is such thatA) F t1/2 B) F t-1/2 C) F v D) F v-1

8. A car of mass m is moving on a level road at a constant speed vmax while facing a resistive force R. If the car slows down to vmax/3, then assuming the engine to be working at the same power, what force F is developing and what is the acceleration a of the car ?A) F = 3R B) F = 2RC) a = 3P/m vmax D) a = 2 P/m vmax

9. The net interacting force F between two particles is related to the distance x between them, as shown in figure. ThenA) potential energy of the system decreases from x1 to x2

B) potential energy of the system increases from x2 to x3

C) potential energy of the system decreases from x3 to x4

D) kinetic energy increases from x1 to x2

10. The potential energy of a particle of mass 1 kg moving in xy plane is given by U = 10 - 4x - 3yThe particle is at rest at (2, 1) at t = 0. ThenA) velocity of particle at t = 1 is 5 m/s B) the particle is at (10m, 7m) at t = 2sC) work done during t = 1s to t = 2s is 75JD) the magnitude of force acting on particle is 5N

11. The figure shows a potential energy function. In the region 0 r R, U(r) is parabolic. In the region r R, U(r) is an inversely proportional function.Identify the correct conservative force function F(r) plot.

(A) (B) (C) (D)12. The potential energy function between two atoms in a diatomic molecule can be

plotted as shown in figure.Mark the correct statement(s):

25

A) There is only one position of equilibriumB) There is two position of equilibriumC) There is only one position of stable equilibriumD) Both the positions are of stable equilibrium

13. A block of mass m is gently placed on a vertical spring of stiffness k. Choose the correct statement related to the mechanical energy E of the system.A) It remains constant B) It decreases C) It increases D) Nothing can be said

14. A spring of stiffness k is pulled by two forces FA and FB as shown in the figure so that the spring remains in equilibrium. Identify the correct statement(s):A) The work done by each force contributes into the increase in potential energy of the springB) The force undergoing larger displacement does positive work and the force undergoing smaller displacement does negative workC) Both the forces perform positive workD) The net work done is equal to the increase in potential energy

15. A particle of mass m is released from a height H on a smooth curved surface which ends into a vertical loop of radius R, as shown in figure.If is the instantaneous angle which the line joining the particle and the centre of the loop makes with the vertical, the identify the correct statement(s) related to the normal reaction N between the block and the surface.A) The maximum value N occurs at = 0 B) The minimum value of N occurs at N =

C) The value of N becomes negative for /2 < <

D) The value of N becomes zero only when > /216. An engine is pulling a train of mass m on a level track at a uniform speed v. The

resistive force offered per unit mass is fA) Power produced by the engine is mfvB) The extra power developed by the engine to maintain a speed v up a gradient of h

in s is

C) The frictional force exerting on the train is mf on the level trackD) None of above is correct

17. A particle of mass 5 kg moving in the x-y plane has its potential energy given by U = (-7x + 24y) J, where x and y are in metre. The particle is initially at origin and has a velocity A) The particle has a speed of 25 ms-1 at t = 4 s B) The particle has an acceleration of 5 ms-2

C) The acceleration of the particle is perpendicular to its initial velocityD) None of the above is correct

18. A particle is moving in a conservative force field from point A to point B. UA and UB

are the potential energies of the particle at points A and B and Wc is the work done in the process of taking the particle from A to BA) Wc = UB - UA B) Wc = UA - UB C) UA > UB D) UB > UA

19. At the position of stable equilibrium

A) 0 only B) = 0 and > 0 C) and D)

None of these20. Choose the correct statement(s) related to the conservative force and potential

energy.A) Potential energy decrease in the direction of conservative forceB) Potential energy increase in the direction of conservative forceC) Conservative force does work by lowering its potential energy

26

D) Conservative force does work by raising its potential energyKEY

1 2 3 4 5 6 7 8 9 10AC BCD AC BD AC BC BD AD ABD ABD11 12 13 14 15 16 17 18 19 20A BC B ACD ABD ABC ABC BC C BC

COMPREHENSION TYPE QUESTIONSPassage I (Q.No: 1 to 7):

The potential energy of two atoms in a diatomic molecule is approximated by U(r) =

, where r is the spacing between atoms and a and b are positive constants.

1. Find the force F(r) on one atom as a function of r:

A) 0 B) C) D)

2. Which is the most appropriate graph U(r) versus r:

3. Which is the most appropriate graph F(r) versus r:

4. Find the equilibrium distance between the two atoms:

A) 2a B) 2a/b C) 2a/5b D)

5. From the above conclusion can we predict about equilibrium state:A) the equilibrium is stable B) the equilibrium is unstableC) the equilibrium may be stable D) the equilibrium may be unstable

6. What minimum energy must be added to the molecule to dissociate it, if the distance between the two atoms is equal to the equilibrium distance found in Q. 4 ?A) b2/a B) 2b2/a C) b2/4a D) 2a/b

7. For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is 1.13 x 10-10 m and the dissociation energy is 1.54 x 10-18 J per molecule. Find the values of a and b:

A) a = 6.67 x 10-138 J-m12 b = 2.08 x 10-60 J m6

B) a = 6.41 x 10-78 J-m6

b = 6.67 x 10-138 J-m12

C) a = 6.67 x 10-138 J-m12 b = 6.41 x 10-78 J m6

D) a = 0b = 6.41 x 10-78 J m6

Passage II (Q.No: 8 to 11) : A cutting tool under microprocessor control has several forces acting on it. One force is , a force in the negative y-direction whose magnitude depend on the

27

A) B) C) D)

A) B) C) D)

position of the tool. The constant is = 2.50 N. Consider the displacement of the tool from the origin to the point x = 3.00 m, y = 3.00 m.

8. Calculate the work done on the tool by if this displacement is along the straight line y = x that connects these two points ?A) 2.50 J B) 500 J C) 50.6 J D) 2 J

9. Calculate the work done on the tool by if the tool is first moved out along the x-axis to the point x = 3.00 m, y = 0 and then moved parallel to the y-axis to x = 3.00 m, y = 3.00 m.A) 67.5 J B) 85 J C) 102 J D) 7.5 J

10. Compare the work done by along these two paths ?A) Work done on x-axis is zeroB) Work done on y-axis is less than on y-axisC) Work done on x-axis is more than on y-axis but not zeroD) Data insufficient

11. What can you predict about ?A) Force is non-conservativeB) Force is conservativeC) Force is neither conservative nor non-conservativeD) Data insufficient to conclude

Passage III (Q.No: 12 to 16): A 1200 kg car is travelling at 7.5 m/s in a northerly direction on an icy road. It crashes into a 8000 kg truck moving in the same direction as the car with a velocity of 3.0 m/s before the collision. The speed of the car after the collision is 3.0 m/s in its original direction.

12. Which of the following is true regarding the relationship between energy and momentum in the passage ?A) The collision is not perfectly elastic, both momentum and energy are not conservedB) The collision is inelastic, kinetic energy is conserved but momentum is notC) The collision is not perfectly elastic, momentum is conserved but total energy is notD) The collision is not perfectly elastic, momentum is conserved but kinetic energy is not

13. What is the velocity of the truck after the collision ?A) 7.5 m/s B) 3.7 m/s C) 3.0 m/s D) 1.1 m/s

14. The car then proceeds to a garage. To get there, the driver turns off onto a smooth road with a coefficient of friction = = 1/4. He then stops for a snack and then tries to drive off. what is the value of frictional force when the force the car exerts is 300 N ?A) 0 N B) 100 N C) 300 N D) 4000 N

15. After leaving the garage, the driver of the car follows the same road and eventually has to go up a hill. How does the frictional force on the car now compare to the value when the car was driving on level ground ?A) No change B) It increased C) It decreasedD) The direction of change depends on the angle of elevation

16. If the car is moving up the hill at 5 m/s and the car is 40m up the hill as shown in the diagram, how much potential energy does the car possess at that point ? (g = 9.8 m/s2).A) 2.40 x 105 J B) 2.40 x 104 JC) 4.95 x 105 J D) 4.95 x 104 J

KEY1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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D A D D A A C C A A A

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MULTIPLE MATCHING TYPE QUESTIONS

1. Match the following:List - I List - IIa) Area under F - S e) Change in KEb) Work energy theorem f) negative of work done to gravitational

forcec) change in PE g) work done by Fd) conservative force h) , where F is conservative force

i) gravitational force 2. Match the following:

List - I List - IIa) KE e) depends on frame of referenceb) work done f) defined for conservative force onlyc) PE g) independent on frame of referenced) spring PE h) same for either compression or elongation for same

distance3. Match the following:

List - I List - IIa) stable equilibrium e) PE in Maxb) unstable equilibrium f) Fnet = 0

c) g) PE is Min

d) h) slope of F-x graph is +ve

4. Match the following:List - I List - IIa) work done by frictional force e) indepent of pathb) work done by electrostatic force f) non-conservativec) work done by gravitational force for closed loop g) depends on pathd) for slowly moving body, wc + wn.c equal to h) define PE

i) zero

KEY1 2 3 4

a-eg, b-e, c-fh, d-i a-e, b-e, c-ef, d-gh a-fg, b-efh, c-g, d-eh a-fg, b-eh, c-i, d-i

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