Kinetics of Particles 2

8/2/2019 Kinetics of Particles 2

1/17

1

Forces

The concept of force is useful because it enables the branches of mechanical science to be

brought together. For example, a knowledge of force required to accelerate a car makes it

possible to decide on the size of the engine and the transmission system. And force can be

treated as a currency between thermodynamics or electro technology or material science.

Newton's Three Laws of Motion

Newton's first law of motion states that if the vector sum of the forces acting on

an object is zero, then the object will remain at rest or remain moving at

constant velocity. If the force exerted on an object is zero, the object does not

necessarily have zero velocity. Without any forces acting on it, including

friction, an object in motion will continue to travel at constant velocity.

The Second Law

Newton's second law relates net force and acceleration. A net force on an

object will accelerate itthat is, change its velocity. The acceleration will be

proportional to the magnitude of the force and in the same direction as the

force. The proportionality constant is the mass, m, of the object.

F = ma

In theInternational System of Units (also known as SI, after the initials of

Systme International), acceleration, a, is measured in meters per second per

second. Mass is measured in kilograms; force, F, in newtons. A newton isdefined as the force necessary to impart to a mass of 1 kg an acceleration of 1

m/sec/sec; this is equivalent to about 0.2248 lb.

A massive object will require a greater force for a given acceleration than a

small, light object. What is remarkable is that mass, which is a measure of the

inertia of an object (inertia is its reluctance to change velocity), is also a

measure of the gravitational attraction that the object exerts on other objects.

It is surprising and profound that the inertial property and the gravitational

property are determined by the same thing. The implication of this

phenomenon is that it is impossible to distinguish at a point whether the pointis in a gravitational field or in an accelerated frame of reference. Einstein

made this one of the cornerstones of his general theory ofrelativity, which is

the currently accepted theory of gravitation.

Friction

Friction acts like a force applied in the direction opposite to an object's

velocity. For dry sliding friction, where no lubrication is present, the friction

force is almost independent of velocity. Also, the friction force does not depend

on the apparent area of contact between an object and the surface upon which

it slides. The actual contact areathat is, the area where the microscopic


2/17

2

bumps on the object and sliding surface are actually touching each otheris

relatively small. As the object moves across the sliding surface, the tiny bumps

on the object and sliding surface collide, and force is required to move the

bumps past each other. The actual contact area depends on the perpendicular

force between the object and sliding surface. Frequently this force is just the

weight of the sliding object. If the object is pushed at an angle to thehorizontal, however, the downward vertical component of the force will, in

effect, add to the weight of the object. The friction force is proportional to the

total perpendicular force.

Where friction is present, Newton's second law is expanded to

The left side of the equation is simply the net effective force. (Acceleration will

be constant in the direction of the effective force). When an object moves

through a liquid, however, the magnitude of the friction depends on thevelocity. For most human-size objects moving in water or air (at subsonic

speeds), the resulting friction is proportional to the square of the speed.

Newton's second law then becomes

The proportionality constant, k, is characteristic of the two materials that are

sliding past each other, and depends on the area of contact between the two

surfaces and the degree of streamlining of the moving object.

The Third Law

Newton's third law of motion states that an object experiences a force because

it is interacting with some other object. The force that object 1 exerts on object

2 must be of the same magnitude but in the opposite direction as the force that

object 2 exerts on object 1. If, for example, a large adult gently shoves away a

child on a skating rink, in addition to the force the adult imparts on the child,

the child imparts an equal but oppositely directed force on the adult. Because

the mass of the adult is larger, however, the acceleration of the adult will be

smaller.

Newton's third law also requires the conservation ofmomentum, or the

product of mass and velocity. For an isolated system, with no external forces

acting on it, the momentum must remain constant. In the example of the adult

and child on the skating rink, their initial velocities are zero, and thus the

initial momentum of the system is zero. During the interaction, internal forces

are at work between adult and child, but net external forces equal zero.

Therefore, the momentum of the system must remain zero. After the adult

pushes the child away, the product of the large mass and small velocity of the

adult must equal the product of the small mass and large velocity of the child.

The momenta are equal in magnitude but opposite in direction, thus adding to

zero.


3/17

3

Another conserved quantity of great importance is angular (rotational)

momentum. The angular momentum of a rotating object depends on its speed

of rotation, its mass, and the distance of the mass from the axis. When a skater

standing on a friction-free point spins faster and faster, angular momentum is

conserved despite the increasing speed. At the start of the spin, the skater's

arms are outstretched. Part of the mass is therefore at a large radius. As theskater's arms are lowered, thus decreasing their distance from the axis of

rotation, the rotational speed must increase in order to maintain constant

angular momentum.1

Newtons Laws of Motion in Summary

(1) Everything continues in its state of rest of rectilinear motion unless disturbed by a force.

(The frame of reference is ignored here!)

(2) The rate of change of momentum ( m x a ) is proportional to the force acting on the bodyand is in the direction of the force (i.e. F = ma)

(3) To each force, there is an equal and opposite reaction.

By Newtons 3rd law, when two objects collide, the sum of momentum before and after must

be equal.

I.e. m1u1 + m2u2 = m1v1 + m2v2

m1 (v1 - u1) = m2 (v2 - u2)

therefore

m

m

v u

v u

speed of mass

speed of mass

2

1

1 1

2 2

1

2

Therefore the 3rd law provides a means of measuring mass and also lead to the principle of

conservation of momentum.

1"Mechanics,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft Corporation.

All rights reserved.


4/17

4

SYSTEM

EQU

ILIBRIUM

(STATICS)

Forces

V

ector

spring

friction

GravitationN

ewton'slawsof

motion

Moment,M=rxF

Axialforce,shear

fo

rce

Bendingmoment,

twistingmoment

Solidmechanicsor

strengtho

fmaterials

kinetics

conservationof

momentum

secondla

w

thirdlaw&

firstlaw

thirdlaw


5/17

5

Spring force

Hookes law

F = k x

therefore

F mx

therefore

kx mx 0

where kis the stiffness of spring.

Friction Force

The friction force between two dry lubricated surface is a quantity which depends on a large

number of factors. (e.g. Ra , clearance, relative velocity), however, consideration of an ideal

case known as Coulomb friction is often regarded as adequate.

The friction force is assumed to take any value

up to a maximum limiting value and

F = N

coefficient of limiting friction

In practice, varies with speed and drops

markedly when sliding begins to occur.

m

natural length

x

F

x

+ve

N

N

FF


6/17

6

Gravitation, the force of attraction between all objects that tends to pull them

toward one another. It is a universalforce, affecting the largest and smallest

objects, all forms ofmatter, andenergy. Gravitation governs the motion of

astronomical bodies. It keeps the moon in orbit around the earth and keeps the

earth and the other planets of the solar system in orbit around the sun. On a

larger scale, it governs the motion of stars and slows the outward expansion ofthe entire universe because of the inward attraction of galaxies to other

galaxies. Typically the term gravitation refers to the force in general, and the

term gravity refers to the earth's gravitational pull.

Gravitation is one of the four fundamental forces of nature, along with

electromagnetism and the weak and strong nuclear forces, which hold together

the particles that make up atoms. Gravitation is by far the weakest of these

forces and, as a result, is not important in the interactions of atoms and

nuclear particles or even of moderate-sized objects, such as people or cars.

Gravitation is important only when very large objects, such as planets, areinvolved. This is true for several reasons. First, the force of gravitation

reaches great distances, while nuclear forces operate only over extremely

short distances and decrease in strength very rapidly as distance increases.

Second, gravitation is always attractive. In contrast, electromagnetic forces

between particles can be repulsive or attractive depending on whether the

particles both have a positive or negative electrical charge, or they have

opposite electrical charges (seeElectricity). These attractive and repulsive

forces tend to cancel each other out, leaving only a weak net force. Gravitation

has no repulsive force and, therefore, no such cancellation or weakening.

The gravitational attraction of objects for one another is the easiest

fundamental force to observe and was the first fundamental force to be

described with a complete mathematical theory by the English physicist and

mathematician Sir Isaac Newton. A more accurate theory called general

relativity was formulated early in the 20th century by the German-born

American physicistAlbert Einstein. Scientists recognize that even this theory is

not correct for describing how gravitation works in certain circumstances, and

they continue to search for an improved theory.

Earth's Gravitation

Gravitation plays a crucial role in most processes on the earth. The ocean

tides are caused by the gravitational attraction of the moon and the sun on the

earth and its oceans. Gravitation drives weather patterns by making cold air

sink and displace less dense warm air, forcing the warm air to rise. The

gravitational pull of the earth on all objects holds the objects to the surface of

the earth. Without it, the spin of the earth would send them floating off into

space.

The gravitational attraction of every bit of matter in the earth for every otherbit of matter amounts to an inward pull that holds the earth together against

the pressure forces tending to push it outward. Similarly, the inward pull of


7/17

7

gravitation holds stars together. When a star's fuel nears depletion, the

processes producing the outward pressure weaken and the inward pull of

gravitation eventually compresses the star to a very compact size (see Star,

Black Hole).

Acceleration

If an object held near the surface of the earth is released, it will fall and

accelerate, or pick up speed, as it descends. This acceleration is caused by

gravity, the force of attraction between the object and the earth. The force of

gravity on an object is also called the object's weight. This force depends on

the object's mass, or the amount of matter in the object. The weight of an

object is equal to the mass of the object multiplied by the acceleration due to

gravity.

A bowling ball that weighs 16 lb is actually being pulled toward the earth with

a force of 16 lb. In the metric system, the bowling ball is pulled toward theearth with a force of 71 newtons (a metric unit of force abbreviated N). The

bowling ball also pulls on the earth with a force of 16 lb (71 N), but the earth

is so massive that it does not move appreciably. In order to hold the bowling

ball up and keep it from falling, a person must exert an upward force of 16 lb

(71 N) on the ball. This upward force acts to oppose the 16 lb (71 N)

downward weight force, leaving a total force of zero. The total force on an

object determines the object's acceleration.

If the pull of gravity is the only force acting on an object, then all objects,

regardless of their weight, size, or shape, will accelerate in the same manner.At the same place on the earth, the 16 lb (71 N) bowling ball and a 500 lb

(2200 N) boulder will fall with the same rate of acceleration. As each second

passes, each object will increase its downward speed by about 9.8 m/sec (32

ft/sec), resulting in an acceleration of 9.8 m/sec/sec (32 ft/sec/sec). In

principle, a rock and a feather both would fall with this acceleration if there

were no other forces acting. In practice, however, air exerts a significant

upward force on the falling feather and makes it fall more slowly.

The mass of an object does not change as it is moved from place to place, but

the acceleration due to gravity, and therefore the object's weight, will changebecause the strength of the earth's gravitational pull is not the same

everywhere. The earth's pull and the acceleration due to gravity decrease as

an object moves farther away from the center of the earth. At an altitude of

4000 miles (6400 km) above the earth's surface, for instance, the bowling ball

would weigh only about 4 lb (18 N). Because of the reduced weight force, the

rate of acceleration of the bowling ball at that altitude would be only one

quarter of the acceleration rate at the surface of the earth. The pull of gravity

on an object also changes slightly with latitude. Because the earth is not

perfectly spherical, but bulges at the equator, the pull of gravity is about 0.5percent stronger at the earth's poles than at the equator.


8/17

8

Early Ideas About Gravitation

The ancient Greek philosophers developed several theories about the force

that caused objects to fall toward the earth. In the 4th century BC, the Greek

philosopherAristotle proposed that all things were made from some

combination of the four elements, earth, air, fire, and water. Objects that were

similar in nature attracted one another, and as a result, objects with moreearth in them were attracted to the earth. Fire, by contrast, was dissimilar and

therefore tended to rise from the earth. Aristotle also developed a cosmology,

that is, a theory describing the universe, that was geocentric, or earth-

centered, with the moon, sun, planets, and stars moving around the earth on

spheres. The Greek philosophers, however, did not propose a connection

between the force behind planetary motion and the force that made objects fall

toward the earth.

At the beginning of the 17th century, the Italian physicist and astronomer

Galileo discovered that all objects fall toward the earth with the sameacceleration, regardless of their weight, size, or shape, when gravity is the

only force acting on them. Galileo also had a theory about the universe, which

he based on the ideas of the Polish astronomerNicolaus Copernicus. In the

mid-16th century, Copernicus had proposed a heliocentric, or sun-centered

system, in which the planets moved in circles around the sun, and Galileo

agreed with this cosmology. However, Galileo believed that the planets moved

in circles because this motion was the natural path of a body with no forces

acting on it. Like the Greek philosophers, he saw no connection between the

force behind planetary motion and gravitation on earth.

In the late 16th and early 17th centuries the heliocentric model of the universe

gained support from observations by the Danish astronomerTycho Brahe, and

his student, the German astronomerJohannes Kepler. These observations,

made without telescopes, were accurate enough to determine that the planets

did not move in circles, as Copernicus had suggested. Kepler calculated that

the orbits had to be ellipses (slightly elongated circles). The invention of the

telescope made even more precise observations possible, and Galileo was one

of the first to use a telescope to study astronomy. In 1609 Galileo observed

that moons orbited the planet Jupiter, a fact that could not reasonably fit intoan earth-centered model of the heavens.

The new heliocentric theory changed scientists' views about the earth's place

in the universe and opened the way for new ideas about the forces behind

planetary motion. However, it was not until the late 17th century that Isaac

Newton developed a theory of gravitation that encompassed both the attraction

of objects on the earth and planetary motion.

Newton's Theory of Gravitation

To develop his theory of gravitation, Newton first had to develop the science offorces and motion calledmechanics. Newton proposed that the natural motion


9/17

9

of an object is motion at a constant speed on a straight line, and that it takes a

force to slow down, speed up, or change the path of an object. Newton also

inventedcalculus, a new branch of mathematics that became an important tool

in the calculations of his theory of gravitation.

Newton proposed his law of gravitation in 1687 and stated that every particle

in the universe attracts every other particle in the universe with a force thatdepends on the product of the two particles' masses divided by the square of

the distance between them. The gravitational force between two objects can be

expressed by the following equation: F= GMm/d2 where F is the gravitational

force, G is a constant known as the universal constant of gravitation, M and m

are the masses of each object, and d is the distance between them. Newton

considered a particle to be an object with a mass that was concentrated in a

small point. If the mass of one or both particles increases, then the attraction

between the two particles increases. For instance, if the mass of one particle is

doubled, the force of attraction between the two particles is doubled. If thedistance between the particles increases, then the attraction decreases as the

square of the distance between them. Doubling the distance between two

particles, for instance, will make the force of attraction one quarter as great as

it was.

According to Newton, the force acts along a line between the two particles. In

the case of two spheres, it acts along the line between their centers. The

attraction between objects with irregular shapes is more complicated. Every

bit of matter in the irregular object attracts every bit of matter in the other

object. A simpler description is possible near the surface of the earth wherethe pull of gravity is approximately uniform in strength and direction. In this

case there is a point in an object (even an irregular object) called the center of

gravity, at which all the force of gravity can be considered to be acting.

Newton's law affects all objects in the universe, from raindrops in the sky to

the planets in the solar system. It is therefore known as the universal law of

gravitation. In order to know the strength of gravitational forces in general,

however, it became necessary to find the value of G, the universal constant of

gravitation. Scientists needed to perform an experiment, but gravitational

forces are very weak between objects found in a common laboratory and thus

hard to observe. In 1798 the English chemist and physicistHenry Cavendish

finally measured G with a very sensitive experiment in which he nearly

eliminated the effects of friction and other forces. The value he found was

6.754 x 10-11 N-m2/kg2close to the currently accepted value of 6.670 x 10-11N-m2/kg2 (a decimal point followed by 10 zeros and then the number 6670).

This value is so small that the force of gravitation between two objects with a

mass of 1 metric ton each, 1 meter from each other, is about 67 millionths of a

newton, or about 15 millionths of a pound.

Gravitation may also be described in a completely different way. A massive

object, such as the earth, may be thought of as producing a condition in space


10/17

10

around it called a gravitationalfield. This field causes objects in space to

experience a force. The gravitational field around the earth, for instance,

produces a downward force on objects near the earth surface. The field

viewpoint is an alternative to the viewpoint that objects can affect each other

across distance. This way of thinking about interactions has proved to be very

important in the development of modern physics.

Planetary Motion

Newton's law of gravitation was the first theory to accurately describe the

motion of objects on the earth as well as the planetary motion that

astronomers had long observed. According to Newton's theory, the

gravitational attraction between the planets and the sun holds the planets in

elliptical orbits around the sun. The earth's moon and moons of other planets

are held in orbitby the attraction between the moons and the planets.

Newton's law led to many new discoveries, the most important of which was

the discovery of the planetNeptune. Scientists had noted unexplainablevariations in the motion of the planetUranus for many years. Using Newton's

law of gravitation, the French astronomer Urbain Leverrier and the British

astronomer John Couch each independently predicted the existence of a more

distant planet that was perturbing the orbit of Uranus. Neptune was

discovered in 1884, in an orbit close to its predicted position.

Problems with Newton's Theory

Scientists used Newton's theory of gravitation successfully for many years.

Several problems began to arise, however, involving motion that did not followthe law of gravitation or Newtonian mechanics. One problem was the observed

and unexplainable deviations in the orbit ofMercury (which could not be

caused by the gravitational pull of another orbiting body).

Another problem with Newton's theory involved reference frames, that is, the

conditions under which an observer measures the motion of an object.

According to Newtonian mechanics, two observers making measurements of

the speed of an object will measure different speeds if the observers are

moving relative to each other. A person on the ground observing a ball that is

on a train passing by will measure the speed of the ball as the same as thespeed of the train. A person on the train observing the ball, however, will

measure the ball's speed as zero. According to the traditional ideas about

space and time, then, there could not be a constant, fundamental speed in the

physical world because all speed is relative. However, near the end of the 19th

century the Scottish physicistJames Clerk Maxwell proposed a complete

theory of electric and magnetic forces that contained just such a constant,

which he called c. This constant speed was 300,000 km/sec (186,000 mi/sec)

and was the speed of electromagnetic waves, including light waves. This

feature of Maxwell's theory caused a crisis in physics because it indicated thatspeed was not always relative.


11/17

11

Albert Einstein resolved this crisis in 1905 with his special theory of relativity.

An important feature of Einstein's new theory was that no particle, and even

no information, could travel faster than the fundamental speed c. In Newton's

gravitation theory, however, information about gravitation moved at infinite

speed. If a star exploded into two parts, for example, the change in

gravitational pull would be felt immediately by a planet in a distant orbitaround the exploded star. According to Einstein's theory, such forces were not

possible.

Though Newton's theory contained several flaws, it is still very practical for

use in everyday life. Even today, it is sufficiently accurate for dealing with

earth-based gravitational effects such as in geology (the study of the formation

of the earth and the processes acting on it), and for most scientific work in

astronomy. Only when examining exotic phenomena such as black holes

(points in space with a gravitational force so strong that not even light can

escape them) or in explaining the big bang (the origin of the universe) isNewton's theory inaccurate or inapplicable.

Einstein's Theory of Relativity

In 1915 Einstein formulated a new theory of gravitation that reconciled the

force of gravitation with the requirements of his theory of special relativity. He

proposed that gravitational effects move at the speed of c. He called this

theory general relativity to distinguish it from special relativity, which only

holds when there is no force of gravitation. General relativity produces

predictions very close to those of Newton's theory in most familiar situations,

such as the moon orbiting the earth. Einstein's theory differed from Newton's

theory, however, in that it described gravitation as a curvature of space and

time.

In Einstein's theory of general relativity, he proposed that space and time may

be united into a single, four-dimensional geometry consisting of 3 space

dimensions and 1 time dimension. In this geometry, called spacetime, the

motions of particles from point to point as time progresses are represented by

curves called world lines. If there is no gravity acting, the most natural lines in

this geometry are straight lines, and they represent particles that are movingalways in the same direction with the same speedthat is, particles that have

no force acting on them. If a particle is acted on by a force, then its world line

will not be straight. Einstein also proposed that the effect of gravitation should

not be represented as the deviation of a world line from straightness, as it

would be for an electrical force. If gravitation is present, it should not be

considered a force. Rather, gravitation changes the most natural world lines

and thereby curves the geometry of spacetime. In a curved geometry, such as

the two-dimensional surface of the earth, there are no straight lines. Instead,

there are special curves called geodesics, an example of which are great

circles around the earth. These special curves are at each point as straight as

possible, and they are the most natural lines in a curved geometry. The effect


12/17

12

of gravitation would be to influence the geodesics in spacetime. Near sources

of gravitation the space is strongly curved and the geodesics behave less and

less like those in flat, uncurved spacetime. In the solar system, for example, the

effect of the sun and the earth is to cause the moon to move on a geodesic that

winds around the geodesic of the earth 12 times a year.

Testing Einstein's Theory

Einstein's theory required verification, but the level of precision in

measurement needed to distinguish between Einstein's theory and Newton's

theory was difficult to achieve in the early 20th century and remains so today.

There were two predictions, however, that could be tested. One involved

deviations in the orbit of Mercury. Astronomers had observed that the ellipse

of Mercury's orbit itself rotatedthat is, the point nearest the sun, called the

perihelion, slowly advanced around the sun. The rate of advancement

predicted by Newton's theory differed slightly from what astronomers had

measured, but Einstein's theory predicted the correct rate.

The second test involved measuring the bending of light as it passed around

the sun. Both Newton's and Einstein's theories predicted that light would be

deflected by gravitation. But the amount of deflection predicted by the two

theories differed. The light to be measured in such a test originates in distant

stars. However, under ordinary conditions the sun's brightness prevents

scientists from observing the light from these stars. This problem disappears

during an eclipse, when the moon blocks the sun's light. In 1919 a special

British expedition took photographs during an eclipse. Scientists measured the

deflection of starlight as it passed by the sun and arrived at numbers that

agreed with Einstein's prediction. Subsequent eclipse observations also have

confirmed Einstein's theory.

Other physicists have proposed relativistic theories of gravitation to compete

with Einstein's. In these competing theories, almost all of which are

geometrical like Einstein's, gravitational effects move at the speed c. They

differ mostly in the mathematical details. Even using the technology of the late

20th century, scientists still find it very difficult to test these theories with

experiments and observations. But Einstein's theory has passed all tests thathave been made so far.

Applications of Einstein's Theory

Einstein's general relativity theory predicts special gravitational conditions.

The Big Bang theory, which describes the origin and early expansion of the

universe, is one conclusion based on Einstein's theory that has been verified in

several independent ways.

Another conclusion suggested by general relativity, as well as other relativistic

theories of gravitation, is that gravitational effects move in waves.Astronomers have observed a loss of energy in a pair of neutron stars (stars

composed of densely packed neutrons) that are orbiting each other. The


13/17

13

astronomers theorize that energy-carrying gravitational waves are radiating

from the pair, depleting the stars of their energy. Very violent astrophysical

events, such as the explosion of stars or the collision of neutron stars, can

produce gravitational waves strong enough that they may eventually be

directly detectable with extremely precise instruments. Astrophysicists are

designing such instruments with the hope that they will be able to detectgravitational waves by the beginning of the 21st century.

Another gravitational effect predicted by general relativity is the existence of

black holes. The idea of a star with a gravitational force so strong that light

cannot escape from its surface can be traced to Newtonian theory. Einstein

modified this idea in his theory of general relativity. Because light cannot

escape from a black hole, for any objecta particle, spacecraft, or wavetoescape, it would have to move past light. But light moves outward at the speed

c. According to relativity, c is the highest attainable speed, so nothing can pass

it. The black holes that Einstein envisioned, then, allow no escape whatsoever.An extension of this argument shows that when gravitation is this strong,

nothing can even stay in the same place, but must move inward. Even the

surface of a star must move inward, and must continue the collapse that

created the strong gravitational force. What remains then is not a star, but a

region of space from which emerges a tremendous gravitational force.

Other Modern Theories

Einstein's theory of gravitation revolutionized 20th-century physics. Another

important revolution that took place was quantum theory. Quantum theory

states that physical interactions, or the exchange of energy, cannot be made

arbitrarily small. There is a minimal interaction that comes in a packet called

the quantum of an interaction. For electromagnetism the quantum is called the

photon. Like the other interactions, gravitation also must be quantized.

Physicists call a quantum of gravitational energy a graviton. In principle,

gravitational waves arriving at the earth would consist of gravitons. In

practice, gravitational waves would consist of apparently continuous streams

of gravitons, and individual gravitons could not be detected.

Einstein's theory did not include quantum effects. For most of the 20th century,theoretical physicists have been unsuccessful in their attempts to formulate a

theory that resembles Einstein's theory but also includes gravitons. Despite the

lack of a complete quantum theory, it is possible to make some partial

predictions about quantized gravitation. In the 1970s, British physicist Stephen

Hawking showed that quantum mechanical processes in the strong

gravitational pull just outside of black holes would create particles and quanta

that move away from the black hole, thereby robbing it of energy.

Theory of Everything

An important trend in modern theoretical physics is to find a theory ofeverything (TOE), in which all four of the fundamental forces are seen to be


14/17

14

really different aspects of the same single universal force. Physicists already

have unified electromagnetism and the weak nuclear force and have made

progress in joining these two forces with the strong nuclear force (see Grand

Unification Theories). However, relativistic gravitation, with its geometric and

mathematically complex character, poses the most difficult challenge. Einstein

spent most of his later years searching for an all-encompassing theory bytrying to make electromagnetism geometrical like gravitation. The modern

approach is to try to make gravitation fit the pattern of the other fundamental

forces. Much of this work involves looking for mathematical patterns. A TOE

is difficult to test using experiments because the effects of a TOE would be

important only in the rarest circumstances.2

Gravitational ForcesBecause the moon has significantly less mass than the earth, the weight of an object on

its surface is only one-sixth the objects weight on the earths surface. This graph shows

how the weight of an object with weight w on earth varies with respect to its position

between the earth and moon. Since the earth and moon pull in opposite directions, there

2"Gravitation,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft Corporation.



15/17

15

is a point, 346,000 km (215,000 mi) from the earth, where the opposite gravitational

forces cancel, and the weight is zero. Microsoft Illustration3

Gravitation

From Issac Newton,

F = G m m

d

1 2

2(1)

F= force of attraction between 2 bodies of masses m1& m2

d= distance between m1 & m2

G = universal gravitational constant

= (6.670 +/- 0.005) x 10-11

m3s2kg

Example :-

mass of Earth = 5.98 x 10

24

kgradius of Earth = 6.368 x 106

m

From (1), for a 1 kg mass on earth surface

F=

6 607 10 5 98 10 1

6 368 10

11 24

62

. .

.

x x x x

x

= 9.8516 N

If this force acts on a unit mass, the acceleration is 9.8516 m/s2

( often called the accelerationdue to gravity and is given symbol g )

Therefore Gravitational force = m x g

However, since the Earth is not a perfect sphere and it also rotates, the declared standard g

value is 9.80665 m/s2

or N/kg

Energy

The quantity calledenergy ties together all branches of physics. In the field of

mechanics, energy must be provided to do work; work is defined as the

product of force and the distance an object moves in the direction of the force.

When a force is exerted on an object but the force does not cause the object to

move, no work is done. Energy and work are both measured in the same

unitsergs, joules, or foot-pounds, for example.

If work is done lifting an object to a greater height, energy has been stored in

the form of gravitational potential energy. Many other forms of energy exist:

electric and magnetic potential energy; kinetic energy; energy stored in

3"Gravitational Forces,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft

Corporation. All rights reserved.


16/17

16

stretched springs, compressed gases, or molecular bonds; thermal energy; and

mass itself. In all transformations from one kind of energy to another, the total

energy is conserved. For instance, if work is done on a rubber ball to raise it,

its gravitational potential energy is increased. If the ball is then dropped, the

gravitational potential energy is transformed to kinetic energy. When the ball

hits the ground, it becomes distorted and thereby creates friction between themolecules of the ball material. This friction is transformed into heat, or

thermal energy.4

Work & Kinetic Energy

Since F = m a

= mdv

dt

therefore in its component form

Fx = mdv

dtm

dx

dt

dv

dxmv

dv

dx

x x

x

x

Similarly, Fy = mvydv

dy

y

Fz = mvz

dv

dz

z

therefore F dx mv dv mv C x x x x 1

2

2

Similarly F dy mv C y y 1

2

2

F dz mv C z z 1

2

2

Therefore F dx F dy F dz mv Cons t x y z 1

2

2tan

where v2 = vx2

+ vy2

+ vz2

therefore Fds = (Fx ii + Fy j + Fz k) (dx i + dy j + dz k)

= (Fx dx + Fy dy + Fz dz)

4"Mechanics,"Microsoft Encarta 97 Encyclopedia. 1993-1996 Microsoft Corporation.



17/17

17

Therefore Fds =1

2

2mv constant

Fds - work done by force Fwhen acting on a particle moving along a given

path.

1

2

2mv - kinetic energy of the particle

dimension of K.E. is kgm2/s

2or (kg m/s

2)m or Nm or J

note :- both work done & energy are scalar.

Power - rate at which work is performed.

I.e. Power = d

dtwork

= ddt

Fds

=d

dtF

ds

dtdt

= Fds

dt= F v (1)

or Power = d

dtK E

d

dtmv. .

1

2

2

Combine with (1), therefore F v = mvdv

dt

= mva

Dimension :- Nm/s or J/s or Watts(W)

Kinetics of Particles 2

Documents

Transcript of Kinetics of Particles 2