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A Matter of Some Gravity
Learning Goal:
To understand Newton's law of gravitation and the distinction between inertial and
gravitational masses.
In this problem, you will practice using Newton's law of gravitation. According to that law, the
magnitude of the gravitational force
between two small particles of masses
and
,
separated by a distance
, is given by
,
where
is the universal gravitational constant, whose numerical value (in SI units) is
.
This formula applies not only to small particles, but also to spherical objects. In fact, the
gravitational force between two uniform spheres is the same as if we concentrated all the mass of
each sphere at its center. Thus, by modeling the Earth and the Moon as uniform spheres, you can
use the particle approximation when calculating the force of gravity between them.
Be careful in using Newton's law to choose the correct value for
. To calculate the force of
gravitational attraction between two uniform spheres, the distance
in the equation for Newton's
law of gravitation is the distance between the centers of the spheres. For instance, if a small object
such as an elephant is located on the surface of the Earth, the radius of the Earth
would be
used in the equation. Note that the force of gravity acting on an object located near the surface of a
planet is often called
weight
.
Also note that in situations involving satellites, you are often given the altitude of the satellite, that
is, the distance from the satellite to the surface of the planet; this is not the distance to be used in the
formula for the law of gravitation.
There is a potentially confusing issue involving mass. Mass is defined as a measure of an object's
inertia, that is, its ability to resist acceleration. Newton's second law demonstrates the relationship
between mass, acceleration, and the net force acting on an object:
. We can now refer to
this measure of inertia more precisely as the
inertial mass
.
On the other hand, the masses of the particles that appear in the expression for the law of gravity
seem to have nothing to do with inertia: Rather, they serve as a measure of the strength of
gravitational interactions. It would be reasonable to call such a property
gravitational mass
.
Does this mean that every object has two different masses? Generally speaking, yes. However, the

MasteringPhysics: Assignment Print View
good news is that according to the latest, highly precise, measurements, the inertial and the
gravitational mass of an object are, in fact, equal to each other; it is an established consensus among
physicists that there is only one mass after all, which is a measure of both the object's inertia and its
ability to engage in gravitational interactions. Note that this consensus, like everything else in
science, is open to possible amendments in the future.

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