Weight and Mass
We have seen that in the universal law of gravitation the crucial quantity
is mass. In popular language mass and weight are often used to mean the same
thing; in reality they are related but quite different things. What we commonly
call weight is really just the gravitational force exerted on an object
of a certain mass. We can illustrate by choosing the Earth as one of the two
masses in the previous illustration of the law of gravitation.
Gravitational Acceleration
As the figure above shows, the weight of an object
at the surface
of the Earth is obtained by multiplying its mass m by the acceleration
due to gravity, g, at that point. The acceleration due
to gravity is approximately the product of the universal gravitational constant
G and the mass of the Earth M, divided by the radius of the
Earth, r, squared. (We assume the Earth to be spherical and neglect
the radius of the object relative to the radius of the Earth in this discussion.)
The measured gravitational acceleration at the Earth's surface is found to be
about 980 cm/second/second or 9.8 m/second/second:
g = acceleration due to gravity = MG/r2 ~ 980
cm/s2 ~ 9.8 m/s2
The quoted value of g is only approximate because it varies a small amount depending
on the exact location on the Earth's surface.
Surface Gravity
The quantity g = MG/r2 for
an object with radius r is often termed the surface gravity
because it is proportional to the gravitational force that would be exerted on
a mass placed at the surface of the object.
The surface gravity is an acceleration, but it is often quoted in units of the Earth's
surface gravity, which makes it just a number without dimension. For example, the
surface gravity of the Moon is 1.62 m/s2, which is
0.17 of the Earth's surface gravity. This means that the force exerted on a test mass
placed on the surface of the Moon would be 0.17 of the force exerted on the same mass
placed at the surface of the Earth.
The gravitational properties such as surface gravity and escape velocity (see the
right panel) for various objects in the Solar System
are summarized in this
table.
Mass and Weight
Mass is a measure of how much material is in an object but weight
is a measure of the gravitational force exerted on that material in a gravitational
field. Thus, mass and weight are proportional to each other, with the acceleration
due to gravity as the proportionality constant. It follows that mass is constant
for an object (actually this is not quite true, but we will save that surprise
for our later discussion of the relativity theory),
but weight depends on the location of the object.
An Example
For
example, if we transported an object of mass m to the surface
of the Moon, the gravitational acceleration would change because the radius
and mass of the Moon both differ from those of the Earth. Our object has
mass m both on the surface of the Earth and on the surface of the Moon.
But it will weigh much less on the surface of the Moon because the
gravitational acceleration there is a factor of six less than at the surface of
the Earth.
Here is a mass and weight
calculator for a mass located at the surface of different
planets in the Solar System and on the surface of the Moon.
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Units for Weight
Since weight is a force, its units are
those of force. The standard metric force unit is
the newton, which is abbreviated N and is defined to be 1 kg m/s2. In the figures and animations on this page we have departed
from our usual use of metric units and employed English System pounds (lb)
to specify weight
because in the United States weights are almost always expressed in that way. The
conversion is 1 lb = 4.448 N, so a 150 lb weight is a 150 x 4.448 = 667.2 N
weight. Dividing 667.2 newtons by the gravitational acceleration of 9.8 m/s2 gives 68 kg for the mass of a 150 lb weight.
Thus, there is a factor of 150 / 68 = 2.2
between the weight in pounds at the Earth's surface and the mass in kilograms.
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