189
As an example, the "twin planets" Uranus and Neptune have nearly the
same mass, but Neptune is about twice as far from the sun as Uranus, so the
sun’s gravitational force on Neptune is about four times smaller.
The forces between heavenly bodies are the same type of
force as terrestrial gravity
OK, but what kind of force was it. It probably wasn’t magnetic, since
magnetic forces have nothing to do with mass. Then came Newton’s great
insight. Lying under an apple tree and looking up at the moon in the sky,
he saw an apple fall. Might not the earth also attract the moon with the
same kind of gravitational force. The moon orbits the earth in the same way
that the planets orbit the sun, so maybe the earth’s force on the falling
apple, the earth’s force on the moon, and the sun’s force on a planet were all
the same type of force.
There was an easy way to test this hypothesis numerically. If it was true,
then we would expect the gravitational forces exerted by the earth to follow
the same F
.
m/r
2
rule as the forces exerted by the sun, but with a different
constant of proportionality appropriate to the earth’s gravitational strength.
The issue arises now of how to define the distance, r, between the earth and
the apple. An apple in England is closer to some parts of the earth than to
others, but suppose we take r to be the distance from the center of the earth
to the apple, i.e. the radius of the earth. (The issue of how to measure r did
not arise in the analysis of the planets’ motions because the sun and planets
are so small compared to the distances separating them.) Calling the
proportionality constant k, we have
F
earth on apple
= k m
apple
/ r
earth
2
F
earth on moon
= k m
moon
/ d
earth-moon
2
.
Newton’s second law says a=F/m, so
a
apple
= k / r
earth
2
a
moon
= k / d
earth-moon
2
.
The Greek astronomer Hipparchus had already found 2000 years before
that the distance from the earth to the moon was about 60 times the radius
of the earth, so if Newton’s hypothesis was right, the acceleration of the
moon would have to be 60
2
=3600 times less than the acceleration of the
falling apple.
Applying a=v
2
/r to the acceleration of the moon yielded an acceleration
that was indeed 3600 times smaller than 9.8 m/s
2
, and Newton was con-
vinced he had unlocked the secret of the mysterious force that kept the
moon and planets in their orbits.
Newton’s law of gravity
The proportionality F
.
m/r
2
for the gravitational force on an object of
mass m only has a consistent proportionality constant for various objects if
they are being acted on by the gravity of the same object. Clearly the sun’s
gravitational strength is far greater than the earth’s, since the planets all orbit
the sun and do not exhibit any very large accelerations caused by the earth
(or by one another). What property of the sun gives it its great gravitational
strength. Its great volume. Its great mass. Its great temperature. Newton
reasoned that if the force was proportional to the mass of the object being
1
60
Section 10.2Newton’s Law of Gravity