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Chapter 14 GRAVITATION

14.1 The World and the Gravitational Force

Claudius Ptolemy (A. D. 2nd century) Nicolaus

Copernicus (1473 ? 1543) Johannes Kepler (1571 ?

1630) Isaac Newton (in 1665) Newtons

theory of celestial mechanics has stood up

remarkably well to the test of centuries and it

remains one of the most accurate and successful

theories in all of physics, that is to say, in

all of science.

aM/aa (0.0027 m/s2)/(9.8 m/s2) 2.8

? 10?4

(6400 km)2/(380,000)2 2.8 ? 10?4

14.2 Newtons Law of Universal Gravitation

14-2-1 Newtons law of universal gravitation

The law of universal gravitation formulated by

Newton states

Every particle attracts every other

particle with a force directly proportional to

the product of their masses and inversely

proportional to the square of the distance

between them.

The constant G is known as the gravitational

constant.

At the surface of the Earth

At high altitude above the surface of the Earth

14-2-2 The vector force The vector form

14.3 Gravitation and the Principle of

Superposition 14-3-1 Principle of Superposition

When more than two bodies are

interacting, the net gravitational force on any

one of them follows the principle of

superposition

14-3-2 The Gravitational constant G

The first laboratory determination of G

from the force between spherical masses at close

distance was done by Henry Cavendish in 1798.

The presently accepted value of G G 6.673

? 10?11 N?m2/kg2 with an uncertainty about ?

0.15. Thus, the mass of the Earth can be

determined.

14.4 Gravitation Near Earths Surface

The magnitude of the gravitational force

acting on a particle of mass m, located at an

external point a distance r from the Earths

center, can then be written as

From Newtons second law, F mag, we obtain

See TABLE 14-1 Variation of ag with Altitude

(page 299).

- The real Earth differs from our model Earth in

three ways - The Earths crust is not uniform. The density of

Earth varies radially as shown in Fig. 14-6 (page

299). There are also local density variations

everywhere. - The Earth is not a sphere. The Earth is

approximately an ellipsoid, flattened at the

poles and bulging at the equator. - The Earth is rotating. The Earth is no longer a

inertial reference frame.

The variation of g with latitude

At the equator

go ? g ?2RE 0.034 m/s2

14.4 Gravitation Inside Earth

There two shell theorems are very useful

in simplifying the analysis of the gravitational

force. Shell Theorem 1 A uniformly

dense spherical shell attracts an external

particle as if all the mass of the shell were

concentrated at its center. Shell Theorem 2

A uniformly dense spherical shell exerts

no gravitational force on a particle located

anywhere inside it.

Therefore, to calculate the gravitational

force on a particle when it is a distance r from

the center of Earth, we may use shell theorems

The gravitational force on the particle is

due only to that portion of the Earth that lies

inside the sphere of radius r, and from shell

theorem 1 it is concluded that we can consider

that mass to be concentrated at the center of the

Earth.

14-6 Gravitational Potential Energy 14-6-1

Calculating the potential energy

The gravitational force is conservative

Thus, we can find the change in the potential

energy of the system as m moves between points a

and b

Take rb ? and Ub 0, we have

The potential energy defined above leads

to the familiar mgy for a small difference in

evaluation y near the surface of the Earth

When y ltlt RE, we can use the binomial expansion

to approximate the last term as (1 x)?1 1 ? x

??? ? 1 ? x, which gives

14-6-2 Escape speed

We can find the escape speed v for the

Earth (or any other bodies) by using conversation

of energy

and solving for v we find

See TABLE 14-2 Some Escape Speeds (page 305).

14-6-3 Potential energy of many-particle systems

We now consider another interpretation

for U(r). Consider two objects, of masses m and

M, separated by an infinitely large distance and

at rest. We take one of the particles (m, for

example) and move it slowly and at constant

velocity toward the other, until the separation

of the two particles is r. Wnet Wext Wgrav

0, Thus the work done by our hand is Wext

? Wgrav ? GMm/r.

We can give this alternative view of the

potential energy The potential energy of

a system of particles is equal to the work done

by an external agent to assemble the system,

starting from the standard reference

configuration.

Consider three bodies of masses m1, m2,

and m3. Let them initially be at rest infinitely

far from one another.

The total work done by the external agent in

assembling the system is

If we wanted to separate the system into three

isolated masses once again, we would have to

supply an amount of energy. This energy is called

binding energy.

14-7 Planets and Satellites Keplers Laws

1. The law of orbits All planets move in

elliptical orbits having the Sun at one focus.

a the semimajor axis e the eccentricity

Aphelion the maximum distance Ra, Ra a(1

e) Perihelion the closest distance Rp, Rp

a(1 ? e) For circular orbits, Ra Rp a.

2. The law of Areas A line joining any planet to

the Sun sweeps out equal areas in equal times.

The rate at which the area is swept out is

Assuming we can regard the more massive body M as

at rest, the angular momentum of the orbiting

body m relative to the origin at the central body

is

Thus

3. The law of periods The square of the period

of any planet about the Sun is proportional to

the cube of the semimajor axis of its orbit. We

take an example from circular orbits

The rotational period T 2?r/v. Thus,

A similar result is obtained for elliptical

orbits, with the radius r replaced by the

semimajor axis a.

Keplers laws also apply to stars orbiting about

each other.

The orbits of the two stars in the binary

system Krüger 60

The orbital motion of Earth is perturbed by

inter-planetary forces, mainly due to Jupiter.

14-8 Energy Considerations in planetary and

satellite motion

Consider again the motion of a body of

mass m about a massive body of mass M. We

consider M to be at rest in an inertial reference

frame, with the body m moving about it in a

circular orbit with tangential speed v and

angular speed ?. The potential energy of the

system is

The kinetic energy is

The total mechanical energy is

For an elliptical orbit

For an elliptical orbit the total energy

is also negative.The quantity r must be taken

equal to the semimajor axis of the ellipse. The

total energy does not depend on the shape of the

ellipse, but only on its overall size.

Incidentally, the angular momenta of these orbits

differ.

E lt 0

E 0

E gt 0

The total energy conservation

The angular momentum conservation

Solving above equations

Focus parameter

Ellipticity

14-9 The Gravitational Field

The point of view of action-at-a-distance

The interaction between the two particles is

direct. The point of view of gravitational field

A particle modifying the space around it in some

way and setting up a gravitational field, then

acts on any other particle, exerting the force of

gravitational attraction on it.

We must determine the gravitational field

established by a given distribution of particles.

Thus, the gravitational field strength at a point

is defined as the gravitational force per unit

mass at that point. In terms of a test mass mo,

And the force on a particle at any point in that

field

14-9 Modern Developments in gravitation 14-9-1

Dark matter

14-9-2 Inertial mass and gravitational mass

The inertial mass and the gravitational

mass of a body are the same.

14-9-3 The principle of equivalence

The similarity between the effects of

gravity and the effects of a suitable accelerated

motion of the frame of reference is called the

Principle of Equivalence.

14-9-3 The principle of equivalence

The similarity between the effects of

gravity and the effects of a suitable accelerated

motion of the frame of reference is called the

Principle of Equivalence.

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14-9-4 The general theory of relativity

Gravitation is due to a curvature (or shape) of

space that is caused by the masses.

- Most significant experimental tests
- Precession of the perihelion of Mercury.
- Bending of light.
- Delay of radar echoes.

Problems

- 14-13 (on page 317),
- 14-17,
- 14-20,
- 14-25,
- 14-44,
- 14-59.