Chapter 9: Einstein and Relativity Theory (319 KB) - D Cassidy Books

3637_CassidyTX_09 6/14/02 12:08 PM Page 431
9.9 RELATIVITY OF MASS 431
9.9 RELATIVITY OF MASS
You saw in Section 3.4 that inertial mass is the property of objects that resists
acceleration when a force is applied. The inertial mass, or simply “the
mass,” is the constant of proportionality between force and acceleration in
Newton’s second law of motion
F net ma.
Therefore a constant force will produce a constant acceleration. So, once
an object is moving, if you keep pushing on it with the same force, it will
keep accelerating, going faster and faster and faster without limit, according
to this formula. Newton’s second law thus contains no speed limit. But
this is inconsistent with the relativity theory, which imposes a speed limit
for objects in space of about 300,000 km/s (186,000 mi/s), the speed of
light. The way out is to amend Newton’s second law. Einstein’s way was to
note that m, the inertial mass, does not stay constant but increases as the
speed increases—as in fact is experimentally observed, for example, for
high-speed elementary particles. When the speed increases, it takes more
and more force to continue the same acceleration—eventually an infinite
force trying to reach the speed of light. Einstein deduced from the two postulates
of special relativity theory that the inertia of a moving object increases
with speed, and it does so in the same way as the time relation in
time dilation. (The derivation is provided in the Student Guide for this chapter.)
Using our familiar square root factor, we can write
m s
m m .
1 /c v
2
2
Here m m is the mass of the object in relative motion, and m s is the mass of
the same object before it starts to move. Often m s is called the “rest mass.”
Similarly to the measurement of time intervals, as an object’s speed increases
the mass as observed from a stationary reference frame also increases.
It would reach an infinite (or undefined) mass if it reached the speed of light.
This is another reason why anything possessing mass cannot actually be
made to attain the speed of light; it would require applying an infinite force
to accelerate it to that speed. By the same argument, entities that do move
at the speed of light, such as light itself, must therefore have zero rest mass.
Following Einstein’s result that the mass of an object increases when it
is in motion relative to a stationary observer, Newton’s equation relating
the force and the acceleration can be written as a more general law
m s
F net
a.
1 /c v
2
2