Simple Nature - Light and Matter

The minimum velocity required for this to happen is calledes-

cape velocity. For speeds above escape velocity, the orbits are

open-ended hyperbolas, rather than repeating elliptical orbits. In

figure i, Pioneer’s hyperbolic trajectory becomes almost indistin-

guishable from a line at large distances from the sun. The motion

slows perceptibly in the first few years after 1974, but later the

speed becomes nearly constant, as shown by the nearly constant

spacing of the dots.

The gravitational field

We got the energy equationU = −Gm 1 m 2 /r by integrating

g∝ 1 /r^2 and then inserting a constant of proportionality to make

the proportionality into an equation. The opposite of an integral is

a derivative, so we can now go backwards and insert a constant of

proportionality ing∝ 1 /r^2 that will be consistent with the energy

equation:

dU=m 2 g 1 dr

g 1 =

1

m 2

dU

dr

=

1

m 2

d

dr

(

−

Gm 1 m 2

r

)

=−Gm 1

d

dr

(

1

r

)

=

Gm 1

r^2

This kind of inverse-square law occurs all the time in nature. For

instance, if you go twice as far away from a lightbulb, you receive

1/4 as much light from it, because as the light spreads out, it is

like an expanding sphere, and a sphere with twice the radius has

four times the surface area. It’s like spreading the same amount of

peanut butter on four pieces of bread instead of one — we have to

spread it thinner.

Discussion Questions

A A bowling ball interacts gravitationally with the earth. Would it

make sense for the gravitational energy to be inversely proportional to the

distance between their surfaces rather than their centers?

2.3.5 The shell theorem

Newton’s great insight was that gravity near the earth’s surface

was the same kind of interaction as the one that kept the planets

from flying away from the sun. He told his niece that the idea

came to him when he saw an apple fall from a tree, which made him

wonder whether the earth might be affecting the apple and the moon

in the same way. Up until now, we’ve generally been dealing with

gravitational interactions between objects that are small compared

to the distances between them, but that assumption doesn’t apply to

102 Chapter 2 Conservation of Energy