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Chapter 17
The Universal Law of Gravitation
104
17
The Universal Law of Gravitation
Consider an object released from rest an entire moon’s diameter above the surface of the
moon.
Suppose you are asked to calculate the speed with which the object hits the
moon
.
This problem typifies the kind of problem in which students use the universal law of
gravitation to get the force exerted on the object by the gravitational field of the moon,
and then mistakenly use one or more of the constant acceleration equations to get the
final velocity.
The problem is: the acceleration is not constant.
The closer the object
gets to the moon, the greater the gravitational force, and hence, the greater the
acceleration of the object.
The mistake lies not in using Newton’s second law to
determine the acceleration of the object at a particular point in space; the mistake lies in
using that one value of acceleration, good for one objecttomoon distance, as if it were
valid on the entire path of the object.
The way to go on a problem like this, is to use
conservation of energy.
Back in chapter 12, where we discussed the nearsurface gravitational field of the earth, we
talked about the fact that any object that has mass creates an invisible forcepermass field in the
region of space around it.
We called it a gravitational field.
Here we talk about it in more detail.
Recall that when we say that an object causes a gravitational field to exist, we mean that it
creates an invisible forcepermass vector at every point in the region of space around itself.
The
effect of the gravitational field on any particle (call it the victim) that finds itself in the region of
space where the gravitational field exists, is to exert a force, on the victim, equal to the forceper
mass vector at the victim’s location, times the mass of the victim.
Now we provide a quantitative discussion of the gravitational field.
(
Quantitative
means,
involving formulas, calculations, and perhaps numbers.
Contrast this with
qualitative
which
means descriptive/conceptual.)
We start with the idealized notion of a point particle of matter.
Being matter, it has mass.
Since it has mass it has a gravitational field in the region around it.
The direction of a particle’s gravitational field at point
P
, a distance
r
away from the particle, is
toward the particle
and the magnitude of the gravitational field is given by
2
m
G
=
g
(171)
where:
G
is the universal gravitational constant:
2
2
11
kg
m
N
10
67
6
⋅
×
=
−
.
G
m
is the mass of the particle, and
is the distance that point
P
is from the particle.
In that point
P
can be any empty (or occupied) point in space whatsoever, this formula gives the
magnitude of the gravitational field of the particle at all points in space.
Equation 171 is the
equation form of Newton’s Universal Law of Gravitation
1
.
1
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 Fall '08
 RABE
 Physics

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