Environmental Engineering Reference
In-Depth Information
4
Angular Momentum
We have seen that the motion of a classical particle is governed by Newton's three
laws, the second of which is the equation of motion
F
m
a
. We have also seen
that, by considering an extended body as a system of particles, the centre of mass
also moves according to
F
=
=
m
a
,where
F
is the sum of the external forces acting
on the body and
m
is the sum of the masses of the constituent particles. In this
way, the dynamical behaviour of an extended body is reduced to the motion of its
centre of mass. However, for an extended body, centre-of-mass motion is only part
of the story. An extended object may also exhibit internal motion. For example,
the centre-of-mass motion of our Solar System tells us nothing about the elliptical
orbits of the individual planets and moons within it and the parabolic motion of
the centre of mass of a ball in flight tells us nothing about the spin of the ball.
The internal motion of an extended system is usually more complicated than the
motion of the centre of mass since it may well be associated with more degrees
of freedom. To deal with this complexity, physicists look for conserved quantities;
properties of the motion that do not change with with time, irrespective of the
internal interactions that occur between the constituent particles. We have seen in
the last chapter that momentum and energy are important conserved quantities. In
this chapter we examine another conserved quantity, which appears both in classical
and quantum physics. That conserved quantity is angular momentum.
4.1 ANGULAR MOMENTUM OF A PARTICLE
To introduce the subject of angular momentum we consider the simplest possible
system, a particle with position vector
r
and momentum
p
. The angular momentum
l
is defined to be
l
=
r
×
p
.
(4.1)
