191

The proportionality to 1/r

2

in Newton’s law of gravity was not entirely

unexpected. Proportionalities to 1/r

2

are found in many other phenomena

in which some effect spreads out from a point. For instance, the intensity of

the light from a candle is proportional to 1/r

2

, because at a distance r from

the candle, the light has to be spread out over the surface of an imaginary

sphere of area 4

p

r

2

. The same is true for the intensity of sound from a

firecracker, or the intensity of gamma radiation emitted by the Chernobyl

reactor. It’s important, however, to realize that this is only an analogy. Force

does not travel through space as sound or light does, and force is not a

substance that can be spread thicker or thinner like butter on toast.

Although several of Newton’s contemporaries had speculated that the

force of gravity might be proportional to 1/r

2

, none of them, even the ones

who had learned Newton’s laws of motion, had had any luck proving that

the resulting orbits would be ellipses, as Kepler had found empirically.

Newton did succeed in proving that elliptical orbits would result from a 1/r

2

force, but we postpone the proof until the end of the next volume of the

textbook because it can be accomplished much more easily using the

concepts of energy and angular momentum.

Newton also predicted that orbits in the shape of hyperbolas should be

possible, and he was right. Some comets, for instance, orbit the sun in very

elongated ellipses, but others pass through the solar system on hyperbolic

paths, never to return. Just as the trajectory of a faster baseball pitch is

flatter than that of a more slowly thrown ball, so the curvature of a planet’s

orbit depends on its speed. A spacecraft can be launched at relatively low

speed, resulting in a circular orbit about the earth, or it can be launched at a

higher speed, giving a more gently curved ellipse that reaches farther from

the earth, or it can be launched at a very high speed which puts it in an

even less curved hyperbolic orbit. As you go very far out on a hyperbola, it

approaches a straight line, i.e. its curvature eventually becomes nearly zero.

Newton also was able to prove that Kepler’s second law (sweeping out

equal areas in equal time intervals) was a logical consequence of his law of

gravity. Newton’s version of the proof is moderately complicated, but the

proof becomes trivial once you understand the concept of angular momen-

tum, which will be covered later in the course. The proof will therefore be

deferred until section 5.7 of book 2.

The conic sections are the curves

made by cutting the surface of an infi-

nite cone with a plane.

An imaginary cannon able to shoot

cannonballs at very high speeds is

placed on top of an imaginary, very

tall mountain that reaches up above

the atmosphere. Depending on the

speed at which the ball is fired, it may

end up in a tightly curved elliptical or-

bit, a, a circular orbit, b, a bigger ellip-

tical orbit, c, or a nearly straight hy-

perbolic orbit, d.

a

b

d

c

hyperbola

ellipse

circle

Section 10.2Newton’s Law of Gravity