chapter 12 gravity

Copyright © 2010 Pearson Education, Inc.

Chapter 12

Gravity


Units of Chapter 12

• Newton’s Law of Universal Gravitation

• Gravitational Attraction of SphericalBodies

• Kepler’s Laws of Orbital Motion

• Gravitational Potential Energy

• Energy Conservation

• Tides


12-1 Newton’s Law of Universal GravitationNewton’s insight:

The force accelerating an apple downward isthe same force that keeps the Moon in its orbit.

Hence, Universal Gravitation.


12-1 Newton’s Law of Universal Gravitation

The gravitational force is always attractive, andpoints along the line connecting the twomasses:

The two forces shown are an action-reactionpair.


12-1 Newton’s Law of Universal Gravitation

G is a very small number; this means that theforce of gravity is negligible unless there is avery large mass involved (such as the Earth).

If an object is being acted upon by severaldifferent gravitational forces, the net force on itis the vector sum of the individual forces.

This is called the principle of superposition.


12-2 Gravitational Attraction of SphericalBodies

Gravitational force between a point mass and asphere: the force is the same as if all the massof the sphere were concentrated at its center.



What about the gravitational force on objects atthe surface of the Earth? The center of the Earth isone Earth radius away, so this is the distance weuse:

Therefore,



The acceleration of gravity decreases slowly withaltitude:


Let M1 = mass of the Earth.22

Mr

GMF

E !"

#$%

&=

Here F = the force the Earth exerts on mass M2. This is theforce known as weight, w.

.222

gMMr

GMw

E

E =!!"

#$$%

&=

2

2m/s 8.9N/kg 8.9 where ===

E

E

r

GMg

Near the surfaceof the Earth

km 6400

kg 1098.5

E

24

E

=

!=

r

M


What is the direction of g?

Note thatm

Fg = is the gravitational force per unit mass.

This is called the gravitational fieldstrength. It is also referred to as theacceleration due to gravity.

What is the direction of w?


Example: What is the weight of a 100 kg astronaut on thesurface of the Earth (force of the Earth on the astronaut)?How about in low Earth orbit? This is an orbit about 300 kmabove the surface of the Earth.

On Earth: N 980== mgw

In low Earth orbit:( )

N 890)(2=!

!"

#$$%

&

+==

hR

GMmrmgw

E

Eo

Their weight is reduced by about 10%.The astronaut is NOT weightless!



Once the altitude becomes comparable to theradius of the Earth, the decrease in theacceleration of gravity is much larger:


12-3 Kepler’s Laws of Orbital Motion

Johannes Kepler made detailed studies of theapparent motions of the planets over many years,and was able to formulate three empirical laws:

1. Planets follow elliptical orbits, with the Sun atone focus of the ellipse.



2. As a planet moves in its orbit, it sweeps out anequal amount of area in an equal amount of time.



3. The period, T, of a planet increases as itsmean distance from the Sun, r, raised to the 3/2power.

This can be shown to be a consequence of theinverse square form of the gravitational force.



A geosynchronous satellite is one whose orbitalperiod is equal to one day. If such a satellite isorbiting above the equator, it will be in a fixedposition with respect to the ground.

These satellites are used for communications andand weather forecasting.



GPS satellites are notin geosynchronousorbits; their orbitperiod is 12 hours.Triangulation ofsignals from severalsatellites allowsprecise location ofobjects on Earth.


12-4 Gravitational Potential Energy

Gravitational potential energy of an object ofmass m a distance r from the Earth’s center:


12-4 Gravitational Potential Energy

Very close to the Earth’s surface, thegravitational potential increases linearly withaltitude:

Gravitational potential energy, just like allother forms of energy, is a scalar. Ittherefore has no components; just a sign.


12-5 Energy Conservation

Total mechanical energy of an object of mass ma distance r from the center of the Earth:

This confirms what we already know – as anobject approaches the Earth, it moves fasterand faster.



Escape speed: the initial upward speed aprojectile must have in order to escapefrom the Earth’s gravity


Additional interesting effects due to gravity

Black holesIf an object is sufficiently massive and sufficiently small, the escape speed will equal orexceed the speed of light – light itself will not be able to escape the surface.

Gravitational lensingLight will be bent by any gravitational field; this can be seen when we view a distantgalaxy beyond a closer galaxy cluster. This is called gravitational lensing, and manyexamples have been found.

TidesWhen two large objects exert gravitational forces on each other, the force on the nearside is larger than the force on the far side, because the near side is closer to the otherobject.

This difference in gravitational force across an object due to its size is called a tidalforce.


Summary of Chapter 12

• Force of gravity between two point masses:

• G is the universal gravitational constant:

• In calculating gravitational forces,spherically symmetric bodies can be replacedby point masses.



• Acceleration of gravity:

• Mass of the Earth:

• Kepler’s laws:

1. Planetary orbits are ellipses, Sun at onefocus

2. Planets sweep out equal area in equal time

3. Square of orbital period is proportional tocube of distance from Sun



• Orbital period:

• Gravitational potential energy:

• U is a scalar, and goes to zero as themasses become infinitely far apart



• Total mechanical energy:

• Escape speed:

• Tidal forces are due to the variations ingravitational force across an extended body

chapter 12 gravity

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