The mass used in Newton's Second
Law, = m_{i} is usually called the inertial mass. This mass is found with
respect to a standard by measuring the respective acceleration of the mass and
the standard when they are made to exert a force on one another. However, when
two masses are weighed on a balance, the measurement records the gravitational
force that is exerted by the earth on each mass that is measured. The mass
determined in this way is called the gravitational mass and it is this
mass that appears in Newton's Law of Universal Gravitation. The assertion
that m_{i} = m_{g} is called the Principle of Equivalence.

There is no obvious reason why the inertial and gravitational masses should be
equal. In fact, if two objects have inertial masses m_{1} and m_{2}, and when
tested by a balance are found to have equal weights w_{1} and w_{2}, then:

w_{1} = w_{2}âá’m_{1}g = m_{2}g

We can infer that m_{1} = m_{2} if and only if g is equal in both cases. That
is, the principle of equivalence holds if the rate of fall due to gravity of
different objects is identical. A great deal of experimental effort has been
made to verifying this hypothesis. It has been determined that the equality
holds to within one part in 10^{12}.

Einstein's Principle of Equivalence

Einstein's General Theory of
Relativity is based on another
principle of equivalence. This asserts that to a local observer (an observer
inside the system), the effects experienced because of an acceleration are
indistinguishable from the effects caused by a gravitational field. If an
astronaut was trapped inside a spaceship with no window, and the spaceship was
accelerating upwards at 9.8 m/sec^{2}, there is no experiment he could do to
determine whether he was still on earth, or accelerating at a remote location in
outer space.

Tides

In addition to the force of gravity from the earth, every object on the earth
must necessarily feel a force from the moon and the sun. However, the earth is
in free fall in relation to both these bodies. Just like the astronaut on the
space shuttle discussed in Gravity Near the
Earth the effects of the pull due to the
sun and earth are "cancelled out" because of the free fall. Yet this
cancellation is not exact; a small net force is exerted by both the moon and the
sun on all objects on the earth. For objects fixed to the surface, this force
is not significant. However, it does act on the oceans, causing them to bulge
toward the moon (or sun) where the moon is closest to the earth and the force is
strongest, and to bulge away where the force is weaker (on the opposite side
from the moon).

As the earth rotates on its axis, the region facing the moon changes, causing
the earth to shift slightly under the oceans. This effect accounts for the
daily rise and fall of the tides.