### Question

What is the difference between the inertial and gravitational mass?

Asked by: Raminder

### Answer

1) Inertial mass. This is mainly defined by
Newton's law, the all-too-famous F = ma, which
states that when a force F is applied to an
object, it will accelerate proportionally, and
that constant of proportion is the mass of that
object. In very concrete terms, to determine the
inertial mass, you apply a force of F Newtons to
an object, measure the acceleration in m/s^{2}, and
F/a will give you the inertial mass m in kilograms.

2) Gravitational mass. This is defined by the force
of gravitation, which states that there is a
gravitational force between any pair of objects,
which is given by

F = G m_{1} m_{2}/r^{2}

where G is the universal gravitational constant,
m_{1} and m_{2} are the masses of the two objects, and
r is the distance between them. This, in effect
defines the gravitational mass of an object.

As it turns out, these two masses are equal to each
other as far as we can measure. Also, the equivalence of these two
masses is why all objects fall at the same rate on
earth.

Answered by: Yasar Safkan, Ph.D. M.I.T., Software Engineer, Istanbul, Turkey

The only difference that we can find between inertial and gravitational mass that we can
find is the method.

**Gravitational mass** is measured by comparing the force of gravity of an unknown mass to the
force of gravity of a known mass. This is typically done with some sort of balance scale.
The beauty of this method is that no matter where, or what planet, you are, the masses will
always balance out because the gravitational acceleration on each object will be the same.
This does break down near supermassive objects such as black holes and neutron stars due to
the high gradient of the gravitational field around such objects.

**Inertial mass** is found by applying a known force to an unknown mass, measuring the
acceleration, and applying Newton's Second Law, m = F/a. This gives as accurate a value for
mass as the accuracy of your measurements. When the astronauts need to be weighed in outer
space, they actually find their inertial mass in a special chair.

The interesting thing is that, physically, no difference has been found between
gravitational and inertial mass. Many experiments have been performed to check the values
and the experiments always agree to within the margin of error for the experiment. Einstein
used the fact that gravitational and inertial mass were equal to begin his Theory of
General Relativity in which he postulated that gravitational mass was the same as inertial
mass and that the acceleration of gravity is a result of a 'valley' or slope in the
space-time continuum that masses 'fell down' much as pennies spiral around a hole in the
common donation toy at your favorite chain store.

To state the answer one more time, there is no difference between gravitational and inertial
mass as far as we know.

Answered by: Matthew Allen, B.S., Physics/Calculus Teacher St. Scholastica Academy