Newton's Laws and His System of the World

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PART C

Newton's Laws & His System of the World

9. Newton's Laws of Motion

10. Rotational Motion

11. Newton’s Law of Universal Gravitation

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Newton's Laws and His System of the World



This subject of the formation of the

three laws of motion and of the law

of gravitation deserves critical

attention.



_{The whole development of thought}

occupied exactly two generations.

It commenced with Galileo and

ended with

Newton's Principia

; and

Newton was born in the year that

Galileo died.



Also the lives of Descartes and

Huygens fall within the period

occupied by these great terminal

figures.

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

As we look into history, it seems that sometimes progress in

a field of learning depended on one person's incisive

formulation of the right problem at the right time. This is so

with the section of mechanics called dynamics, the science

that deals with the efects of forces on moving bodies.



The person was Isaac Newton, and the formulation was that

of the concepts force and mass, expounded in three

interconnected statements that have come to be called

Newton's three laws of motion.



But also, with the help of Newton's law of universal

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Newton's Laws of Motion

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9.1 Science in the 17

th

Century

 _{Covering less than half of the 17}th century and only the physical sciences, will show

the justification for the term "the century of genius" :

 _{The work on vacuums and pneumatics by}_{Torricelli, Pascal, von Guericke, Boyle,} and Mariotte;

 Descartes‘ great studies on analytical geometry and on optics;

 _Huygens_{' work in astronomy and on centripetal force, his perfection of the}

pendulum clock, and his book on light;

 _{The establishment of the laws of collisions by}_{Wallis, Wren,}_and_Huygens_;

 _Newton's_{work on optics, including the interpretation of the solar spectrum, and his}

invention of calculus almost simultaneously with Leibniz;

 _{The opening of the famous Greenwich observatory; and}  _Hooke's_{work, including that on elasticity.}

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

One aspect is that both craftsmen and men of leisure

and money begin to turn to science, the one group

for improvement of methods and products and the

other for a new and exciting hobby-as amateurs (in

the original sense, as lovers of the subject).



But availability of money and time, the need for

science, and the presence of interest and

organizations do not alone explain or sustain such a

thriving enterprise.



Even more important ingredients are

able

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9.2 A Short Sketch of Newton's Life

 _{Isaac Newton was born on Christmas day in 1642, in the small village of}

Woolsthorpe in Lincolnshire, England.

 _{He was a quiet farm boy who, like young Galileo, loved to build and tinker with}

mechanical gadgets and seemed to have a secret liking for mathematics.

 _{Through the fortunate intervention of an uncle he was allowed to go to Trinity}

College, Cambridge University, in 1661 (where he appears to have initially enrolled in the study of mathematics as applied to astrology!).

 _{He proved an eager and excellent student. By 1666, at age 24, he had quietly}

made spectacular discoveries in mathematics (binomial theorem, diferential calculus), optics (theory of colors), and mechanics.

 Referring to this period, Newton once wrote:

 _{And the same year}_I_{began to think of gravity extending to the orb of the Moon, and} . . . from Kepler's Rule [third law] . . . I deduced that the forces which keep the Planets in their orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and pIinded Mathematics and Philosophy more than at any time since.

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

From his descriptions we may conclude that during those

years of the plague, having left Cambridge for the time to

study in isolation at his home in Woolsthorpe, Newton had

developed a clear idea of the first two laws of motion and

of the formula for centripetal acceleration, although he did

not announce the latter until many years after Huygens'

equivalent statement.



After his return to Cambridge, he did such creditable work

that he followed his teacher as professor of mathematics.

He lectured and contributed papers to the Royal Society,

at first particularly on optics. His Theory of Light and

Colors, when finally published, involved him in so long and

bitter a controversy with rivals that the shy and

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

Newton now concentrated most on an extension

of his early eforts in celestial mechanics the study

of planetary motions as a problem of physics.



In1684 the devoted friend Halley came to ask his

advice in a dispute with Wren and Hooke as to the

force that would have to act on a body executing

motion along an ellipse in accord with Kepler's

laws; Newton had some time before found the

rigorous solution to this problem ("and much other

matter") . Halley persuaded his reluctant friend to

publish the work, which touched on one of the

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

In less than two years of incredible labors the

Principia was ready for the printer; the

publication of the volume (divided into three

"Books") in 1687 established Newton almost

at once as one of the greatest thinkers in

history.



A few years afterward, Newton, who had

always been in delicate health, appears to

have had what we would now call a nervous

breakdown.



On recovering, and for the next 35 years until

his death in 1727, he made no major new

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9.3 Newton's Principia



In the original preface to Newton's we find a clear outline:



_{Since the ancients considered mechanics to be of the greatest}

importance in the investigation of nature and science, and since the

moderns have undertaken to reduce the phenomena of nature to

mathematical laws, it has seemed best in this treatise to concentrate

on mathematics as it relates to natural philosophy [we would say

"physical science"] . . .



_{For the basic problem of philosophy seems to be to discover the forces}

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

What a prospect! The work begins with a set of definitions: in this case,

mass, momentum, inertia, force, and centripetal force. Then follows a

section on absolute and relative space, time, and motion.



Mass

 Newton was very successful in using the concept of mass, but less so in

clarifying its meaning. He states that mass or "quantity of matter“ is "the product of density and bulk" (bulk = volume). But what is density? Later on in the

Principia, he defines density as the ratio of "inertia“ to bulk; yet at the beginning, he had defined inertia as proportional to mass.



Thus a modem text or encyclopedia typically states that mass is simply an

undefined concept that cannot be defined in terms of anything more

fundamental, but must be understood operationally as the entity that

relates two observable quantities, force and acceleration. Newton's second

law, to be introduced below, can be regarded as a definition of mass.



What is more important is that Newton clearly established the modem

distinction between mass and weight-the former being an inherent

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

Time.

_{Newton writes:}

example, an hour, a day, a month, a year-is commonly used instead of true time.



_Space.

_{Newton continues:}

 Absolute space, of its own nature without reference to anything external, always

remains homogeneous and immovable.

 Relative space is any movable measure or dimension of this absolute space; such

a measure or dimension is determined by our senses from the situation of the space with respect to bodies . . .

 _{Since these parts of space cannot be seen and cannot be distinguished from one}

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

In the introductory section of the Principia, Newton stated

his famous three laws of motion and the principles of

composition of vectors (for example, of forces and of

velocities).

Book 1

, titled "

The Motion of Bodies

," applies

these laws to problems of interest in theoretical astronomy:

the determination of the orbit described by one

body around another, when they are assumed to interact

according to various force laws, and mathematical theorems

concerning the summation of gravitational forces exerted by

diferent parts of the same body on another body inside or

outside the first one.



Book 2

, on “

The motion of bodies

in resisting

mediums

”



Along with this there are a number of theorems and

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

Book 3

, "

The System of the World

," is the

culmination: It makes use of the general results

derived in Book 1 to explain the motions of the

planets and other gravitational phenomena such as

the tides. It begins with a remarkable passage on

"Rules of Reasoning in Philosophy."



The four rules, refecting the profound faith in the

uniformity of all nature, are intended to guide

scientists in making hypotheses, and in that function

they are still up to date.

The frst

has been called a

principle of parsimony, and

the second

and

third

,

principles of unity.

The fourth

is a faith without

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

_{Rule 1}



No more causes of natural things should be admitted than are both true

and sufcient to explain their phenomena.



Rule 2



_{Therefore, the causes assigned to natural efects of the same kind must}

be, so far as possible, the same.



Rule 3



_{Those qualities of bodies that cannot be intended and remitted [that is,}

qualities that cannot be increased and diminished] and that belong to

all bodies on which experiments

can be made should be taken as qualities of all bodies universally.



Rule 4



In experimental philosophy. propositions gathered from phenomena

by induction should be considered exactly or very nearly true

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9.4 Newton's First Law of Motion



We may phrase Newton's first law of motion, or

law of inertia

, as

follows:

Every material body persists in its state of rest or of

uniform, unaccelerated motion in a straight line, if and only if it

is not acted upon by a net (that is, unbalanced external) force

.



_{The essence is this: If you see a moving body deviating from a straight}

line, or accelerating in any way, then you must assume that a net force

(of whatever kind) is acting on the body; here is the criterion for

recognizing qualitatively the presence of an unbalanced force.



There is implied only the definition of force as the "cause" of change of

velocity, a definition we already alluded to in our discussion of Galileo's

work. We recall that the Aristotelian scholastics had a rather diferent

view in this matter; they held that force was also the cause of uniform

(unaccelerated) motion.



_{It has been made plain that a net force must be supplied to change a}

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

Problem 9. 1 .

Explain in terms of Newton's

first law of motion the common experience of

lurching forward when a moving train

suddenly decelerates and stops. Explain what

happens to the passengers in a car that

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9.5 Newton's Second Law of Motion



In Newton's own formulation of the second law, he

states that

the force acting on a body is equal

to the change of its quantity of motion,

where "quantity of motion" (later called

"momentum ") is defned as the product of

mass and velocity.



It is customary to begin by stating the second law

of motion in the form:

The net (external

unbalanced) force acting on a material body

is directly proportional to, and in the same

direction as, its acceleration.

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9.5 Newton's Second Law of Motion



The constant so defined is a measure of the

body's inertia, for clearly a large ratio of

F

_net

to

a means that a large force is needed to

produce a desired acceleration, which is just

what we expect to find true for large, bulky

objects



If we now symbolize the constant in Eq. (9.1 ),

the measure of inertia, by a letter of its own,

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

In summary:

Newton's second law, in

conjunction with the essentially arbitrary

choice of one standard of mass, conveniently

fixes the unit of force, permits the calibration

of balances, and gives us an operational

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9.6 Standard of Mass



It will readily be appreciated that the Standard of Mass that represents 1

kilogram, though essentially arbitrary, has been chosen with care. For

scientific work, 1/1000 of 1 kilogram, equal to 1 gram, was originally

defined as the mass of a quantity of 1 cubic centimeter (1 cm

3

) of distilled

water at 4°C. This decision, dating from the late eighteenth century, is

rather inconvenient in practice. Although standardizing on the basis of a

certain amount of water has the important advantage that it permits

cheap and easy reproduction of the standard anywhere on earth, there are

obvious experimental difculties owing to the efects of evaporation, the

additional inertia of the necessary containers, relatively poor accuracy for

measuring volumes, and so on.



Therefore, it became accepted custom to use as the standard of mass a

certain piece of precious metal, a cylinder of platinum alloy kept under

careful guard at the Bureau lnternationale des Poids et Mesures at Sevres,

a suburb of Paris (alongside what for a long time was defined as the

standard of length, a metal bar regarded as representing the distance of 1

meter) . For use elsewhere, accurate replicas of this international standard

of mass have been deposited at the various bureaus of standards

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

_{Fig. 9.3. Standard kilogram, a}

platinum-iridium cylinder

constructed in 1878, held at

Sevres, France, together with a

standard mete r.



Again, it was a matter

of convenience and

accuracy to make this

metal block have an

inertia 1000 times

that of a I -g mass;

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9.7 Weight

 _{Objects can be acted on by all}_{kinds of forces}_;  _{by a push from the hand;}

 by the pull on a string or spring balance attached to the object;  _{by a collision with another object;}

 _{by a magnetic attraction if the object is made of iron or other susceptible materials;}  _{by the action of electric charges;}

 _{by the gravitational attraction that the earth exerts on bodies.}

 _{But no matter what the origin or cause of the force, and no matter where in the}

universe it happens, its efect is always given by the same equation, Fnet = ma.

Newton's second law is so powerful precisely because it is so general, and because we can apply it even though at this stage we may be completely at a loss to

understand exactly why and how a particular force (like magnetism or gravity) should act on a body.

 If the net force is, in fact, of magnetic origin, we might write F_mag = ma;  _{if electric,}_F_el₌_ma_{; and so forth.}

 Along the same line, we shall use the symbol F_grav when the particular force involved

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

Presupposed throughout the previous paragraphs was

some accurate method for measuring

F

grav

.We might

simply drop the object, allowing

F

grav

to pull on the object

and to accelerate it in free fall, and then find the

magnitude of

F

grav

by the relation.



Happily there is another method that is easier and more

direct. We need only our previously calibrated spring

balance; from it we hang the body for which

F

grav

is to be

determined, and then wait until equilibrium is

established. Now we do not allow F

grav

downward to be

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Fig. 9.4. Weighing

with a spring

balance.



the pointer comes to rest-say on

the 5-newton reading-then we

know (by Newton's first law) that

the upward pull of the spring,

F

bal

, just counter balances the

downward pull of

F

grav

on the

object. The net force on the body

is zero. While oppositely

directed, these two forces on the

same object are numerically

equal, and therefore F

grav'

the

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

In summary

, the dynamic

method of measuring

weights by

F

grav

=

m

x

g

involves a prior

determination of mass

m

and also a measurement of

g

. Now, while g is constant

for all types of objects at a

given locality and may for

most purposes be taken as

9.80 m/sec

2

or 32.2 ft/sec

2

on the surface of the earth,

the exact value is

measurably diferent at

diferent localities.

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9.8 The Equal-Arm Balance



_{Before we leave the crucial-and initially perhaps}

troublesome--concept of mass,

we must mention a third way of

measuring the mass of objects

, in everyday practice by far

the most favored and accurate method.



By way of review, recall first that we need the essentially

arbitrary standard. Once this has been generally agreed on,

we can calibrate a spring balance by using it to give the

standard measurable accelerations on a smooth horizontal

plane.



Then the calibrated balance can serve to determine other,

unknown masses, either :

(a) by a new observation of pull and resulting acceleration on the

horizontal plane, or

(b) by measuring on the balance

F

grav

for the object in question and

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Fig. 9.5. Weighing

with an equal-arm

balance.

 _{To methods (a) and (b) we now add method (c),}

known of course as "weighing" on an equal arm balance-at first glance seemingly simple and straightforward, but in fact conceptually most deceptive.

 We place the unknown mass m_x on one pan (Fig.

9.5) and add a sufcient quantity of calibrated and marked auxiliary-standard masses on the other side to keep the beam horizontal.

 _{When this point is reached, the gravitational pull}

on the unknown mass, namely mx g, is

counterbalanced by the weight of the standards, m,g. (Note that we assume here the sufciently verified experimental result concerning the equality of gravitational accelerations for all masses at one locality. )

 But if m_xg = m_sg, then m_x = m_s. Counting up the

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9.9 Inertial and Gravitational Mass

 _{We see that case (a) on one hand and, (b) and (c) on the other measure two}

entirely diferent attributes of matter, to which we may assign the terms inertial mass and gravitational mass, respectively.

 For practical purposes we shall make little distinction between the two types of

mass. But in order to remind ourselves how essential and rewarding it may be to keep a clear notion of the operational significance of scientific concepts, and that, historically, considerable consequences may follow from a reconsideration of long-established facts in a new light, there are these words from Albert

Einstein's and Leopold Infeld's book The Evolution of Physics:

 _{Is this identity of the two kinds of mass purely accidental, or does it have a deeper}

significance? The answer, from the point of view of classical physics, is: the identity of the two masses is accidental and no deeper significance should be attached to it. The answer of modern physics is just the opposite: the identity of the two masses is

fundamental and forms a new and essential clue leading to a more profound

understanding. This was, in fact, one of the most important clues from which the so-called general theory of relativity was developed.

 _{A mystery story seems inferior if it explains strange events as accidents . It is certainly}

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9.10 Examples and Applications of

Newton's Second Law of Motion



Examples 1.

An object of mass m (measured in kg)

hangs from a calibrated spring balance in an elevator

(Fig. 9 . 6 ) . The whole assembly moves upward with a

known acceleration a (measured in m/sec

2

). What is the

reading on the balance?



Solution:

As long as the elevator is stationary, the

upward pull of the balance, and hence its reading

F

I

,

will be equal in magnitude to the weight mg of the

object. The same is true if the elevator is moving up or

down with constant speed, which is also a condition of

equilibrium and of cancellation of all forces acting on

m

. However, in order to accelerate upward with

a

m/sec

2

,

m

must be acted on by a net force of

ma

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Fig. 9. 6. Weight of

an object suspended

in an elevator

so



The reading F

_I

will be larger

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Fig. 9. 7.

Atwood's

machine .



Examples 2.

A string, thrown over a very light

and frictionless pulley, has attached at its ends

the two known masses m

1

and m

2

, as shown in

Fig. 9.7. Find the magnitude of the acceleration

of the masses.



Solution:

In this arrangement, called Atwood's

machine, after the eighteenth-century British

physicist who originated it, the net external

force on the system of bodies is m

2

g = -m

1

g

(again assuming m

2

to be larger than m

1

) The

total mass being accelerated is m

1

+ m

2

, and,

consequently,



which we may solve with the known values of

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9.11 Newton's Third Law of Motion

 _{Newton's frst law}_{defined the force concept qualitatively, and}_{the second} law quantified the concept while at the same time providing a meaning for the idea of mass. To these, Newton added another highly original and important law of motion, the third law, which completes the general characterization of the concept of force by explaining, in essence, that each existing force on one body has its mirror-image twin on another body.

 _{In Newton's words,}

 _{To any action there is always an opposite and equal reaction; in other words, the}

actions of two bodies upon each other are always equal and always opposite in

direction: Whatever presses or draws something else is pressed or drawn just as much by it.

 _{If anyone presses a stone with a finger, the finger is also pressed by the stone.}

 _{If a horse draws a stone tied to a rope, the horse will (so to speak) also be drawn back}

equally towards the stone, for the rope, stretched out at both ends, will urge the horse toward the stone and the stone toward the horse by one and the same endeavor to go slack and will impede the forward motion of the one as much as it promotes the

forward motion of the other.

 _(Principia)

 _{To emphasize all these points, we might rephrase the third law of motion as}

follows: Whenever two bodies A and B interact so that body A

experiences a force (whether by contact, by gravity, by magnetic

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9.12 Examples and Applications of Newton's

Third Law

 _{Example 1.}_{The simplest case concerns a box (body A)}

standing on the earth (body B). Let us identify the forces that act on each. Probably the one that comes to mind first is the weight of the box, Fgrav' We name it here F1A and enter it as a

vertically downward arrow, "anchored" to the box A at its center of gravity (see Fig. 9.10a).

 _{The reaction that must exist simultaneously with this pull of}

the earth on the box is the pull of the box on the earth, equally large (by the third law) and entered as a vertical,

upward arrow, F1B at the center of the earth in Fig. 9.10b. This

completely fulfills the third law. However, as the second law informs us, if this were the complete scheme of force, the box should fall down while the earth accelerates up.

 _{This is indeed what can and does happen while the box drops}

to the earth, settles in the sand, or compresses the stones beneath it. In short, the two bodies do move toward each other until enough mutual elastic forces are built up to balance the previous set.

 _{Specifically, the earth provides an upward push on the box at}

the surface of contact, as shown in Fig. 9.10c by F2A an

upward arrow "attached" to the box, while, by the law now under discussion, there exists also an equal, oppositely directed force on the ground, indicated by F2B in Fig. 9.10d.

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

There are now two forces on each body.



Equilibrium is achieved by the equality in magnitude of

F

_1A

and F

_2A

on A, and by F

_1B

and F

_2B

on B. But beware!

F

_1A

and F

_2A

are not to be interpreted as action and

reaction, nor are F

_1B

and F

_2B

.



The reaction to F

_1A

is F

_1B

, and the reaction to F

_2A

is F

_2B

.

Furthermore, F

₁

and F

₂

are by nature entirely diferent

sets of forces the one gravitational and the other

elastic.



In short, F

_1A

and F

_1B

are equally large by Newton's third

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

Example 2.

The sketch in Fig. 9. 11 involves a

system of four elements-a horizontal stretch

of earth E on which a recalcitrant beast of

burden B is being pulled by its owner M by

means of a rope R.

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

Follow these four force couples:

F

_1E

is the push experienced by the earth,

communicated to it by the man's heels (essentially by static friction).



The reaction to

F

_1E

is the equally large force

F

_1M

exerted on the man by the



_{In equilibrium, the separate forces on each of the four objects balance; but}

if equilibrium does not exist, if the man succeeds in increasing the donkey's

speed to the left, then

F

3B

-

F

4B

=

m

beast

x

a

, and similarly for the other

members of the system.



_{And whether there is equilibrium or not, any "action" force is equal and}

opposite to its " reaction."



The whole point may be phrased this way:

 By the third law, the forces F_IE and F_1M are equal; similarly, F_2M and F_2R are equal.  _{But the third law says nothing whatever about the relationship of}_F_1M_to_F_2M_two

forces arranged to act on the same body by virtue of the man's decision to pull on a rope, not by any necessity or law of physics.

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CHAPTER 10 : Rotational Motion



In the previous chapters we first acquainted ourselves with the

description of uniformly accelerated motions along a straight line, and in

particular with that historically so important case of free fall. Next came

general projectile motion, an example of motion in a plane considered

as the superposition of two simple motions. Then we turned to a

consideration of the forces needed to accelerate bodies along a straight

line.



But there exists in nature another type of behavior, not amenable to

discussion in the terms that we have used so far, and that is rotational

motion, the motion of an object in a plane and around a center, acted on

by a force that continually changes its direction of action.



_{This topic subsumes the movement of planets, fywheels, and}

elementary particles in cyclotrons. We shall follow the same pattern as

before: concentrating on a simple case of this type, namely, circular

motion.



We shall first discuss the kinematics of rotation without regard to the

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10.1 Kinematics of Uniform Circular Motion



Consider :



_{a point whirling with constant speed in a circular path about a center}

O;



_{the point may be a spot on a record turntable, or}



a place on our rotating globe, or,



to a good approximation, the planet Venus in its path round the sun.



Before we can investigate this motion, we must be able to

describe it. How shall we do so with economy and precision?

Some simple new concepts are needed:

a) The frequency of rotation

is the number of revolutions

made per second (letter symbol n), expressed in I/sec (or sec

-t

).

A wheel that revolves 10 times per second therefore has a

frequency of

n

= 10 sec

-1

While useful and necessary, the

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b) Next, we define

the concept period of rotation

(symbol T) as

the number of seconds needed per complete revolution, exactly

the reciprocal of n, and expressed in units of seconds.

The wheel would consequently have a period of rotation of 0.1 sec.

c)

An angular

measure is required. The angle a swept through by a

point going from P

1

to P

2

(Fig. 10.1) can, of course, be measured in

degrees, but it is more convenient in many problems to express



by the defining equation

Fig. 10. 1. Definition of an angle

in radian measure:



=

s

/

r.

where

s

is the length of the arc

and r is the radius of the circle.

This ratio of arc to radius is a

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d) We now inquire into

the

velocity of a particle

executing uniform circular

motion. The word "uniform"

means, of course, that the rate

of spin (the speed

s

/

t

) does not

change. Nevertheless, for

future reference, it is well to

remember that the velocity

vector associated with the

rotating point does change in

direction from one instant to

the next, although its

magnitude, represented by the

length of the arrows in Fig.

10.2, is here constant.

 _{Let us now concentrate entirely on}

the magnitude of the velocity, the speed, given again by the ratio of distance covered to time taken; if we know the period or frequency of the motion and the distance r from the spot to the center of the circle, v is directly found (usually in cm/sec) by realizing that s = 2 r if t = T, that is,

e) The quantity  defined in this last

equation refers to the magnitude of the tangential or linear velocity, i.e., to the velocity of the point along the direction of its path. Analogous to this time rate of change of distance

stands the powerful concept angular velocity [symbolized by the Greek letter  (omega)], which is the time

rate of change of angle. By definition, for the type of motion we are here considering,

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

If we happen to know

n

or

T

, we can find the

magnitude of the angular velocity from the

fact that



= 2



if

t

=

T

, or



The formal relation between

_

and

v

is

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10.2 Centripetal Acceleration



In the previous section that motion with constant

speed around a circle implies that the velocity vector is

continually changing in direction though not in

magnitude.

According to Newton's laws of motion, a force must be

acting on a body whose velocity vector is changing in

any way, since if there were no net force it would

continue to move at constant velocity in a straight line.



And if there is a force, there must be an acceleration.

So when in circular motion with constant speed, a body

is in the seemingly paradoxical situation of being

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

In the case of circular motion, the change in direction of

the velocity vector was shown in Fig. 10.2; we now need

to analyze this change in a little more detail (Fig. 10.3).



The vector labeled "v at P

₂

" is the resultant of two other

vectors, which must be added together: the vector "v at

P

I

" and the vector "



v

" which represents the change in

velocity that occurs during the time interval 

t

as the

body moves along the circle from P

I

to P

2

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

As can be seen from the· diagram, the vector

_

v

is

directed toward the center of the circle. The acceleration

is defined as the change in velocity divided by the time

interval, in the limit as the time interval becomes very

small. In symbols:



Acceleration is also a vector, directed toward the center

of the circle; hence it is called the

centripetal

acceleration

("centripetal" means "seeking the center”)

.



The corresponding force that must be acting on the body

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

Unfortunately the ordinary language that we use to

describe motion causes confusion at this point. We are

accustomed to hearing talk about

"centrifugal"

force

, a force that is said to act on a whirling body in

a direction away from the center.



We shall not use this term, because there is no such

force, despite the illusion. If you tie a weight on a

string and twirl it around your head (Fig. 1 0.4)

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