Environmental Engineering Reference
In-Depth Information
F
21
F
12
r
1
r
2
O
Figure 4.2
A mutual interaction between two particles that gives rise to a net torque.
F
12
and
F
21
on each other. Although these forces are equal in magnitude and have
opposite directions (consistent with Newton's Third Law) that is not sufficient to
argue that the net torque is also zero. Thus, we cannot rely on Newton's laws
to make a statement about the sum of internal torques. Nevertheless, it is an
experimental fact that these torques sum to zero and it is at this stage that new
physics enters our development. We therefore demand that
N
N
r
j
×
F
jk
=
0
,
(4.17)
j
=
1
k
=
1
which then gives for the rate of change of angular momentum:
N
d
L
d
t
=
F
(e)
j
r
j
×
=
τ
,
(4.18)
=
1
j
where the total torque
is equal to the total torque due only to external forces
acting on the body. The physics of Eq. (4.17) is not hard to understand. If it did
not hold then the angular momentum of a body could change even if no external
torques act upon it and that would lead to the bizarre result that isolated bodies
could spontaneously start to rotate. We have thus arrived at a statement of the
principle of conservation of angular momentum:
τ
In the absence of external torques, the total angular momentum of a system is a
conserved quantity.
4.3 ANGULAR MOMENTUM AND ROTATION ABOUT A FIXED AXIS
We have so far succeeded in finding an equation of motion that relates the
angular momentum to the net external torque. Our task in this section is to make
more explicit the link between the angular momentum and the spin of the rotating
body. To help simplify matters we will assert that the system of particles constitutes
a rigid body. By this we mean that the relative positions of the particles that make
up the body are fixed. Furthermore, we will make the restriction that the body is
rotating about an axis that has a fixed direction in space. Examples of rotation of
