466 Chapter 16Vectors
where I 1 = 1 mr
2is the moment of inertia about the axis of rotation. The relation
between land yis less simple than that given by (16.61) for arbitrary motion of a
particle or for the rotation of a rigid body. The general relation is
l 1 = 1 I
ω
(16.62)where Iis a quantity with nine components called the moment of inertia tensor (or
matrix), and is discussed in Example 19.15.
0 Exercise 44
Conservation of angular momentum
As discussed in Example 16.9, Newton’s second law of motion can be written as
Taking the cross product of rwith both sides of this equation gives
The left side is the torque Tacting on the system (Example 16.16). The right side is,
using the product rule of differentiation (valid for both scalar and vector products),
But is parallel to p, and their vector product is zero. Therefore, sincer 1 z 1 p 1 = 1 l,
(16.63)
This form of Newton’s second law, that the rate of change of angular momentum of a
system is equal to the applied torque, shows that the angular momentum of a system
is constant in the absence of external torques.
0 Exercises 45– 47
16.7 Scalar and vector fields
A function of the coordinates of a point in space is called a function of position or
field. A scalarfunction of position, a scalar field,
f 1 = 1 f(r) 1 = 1 f(x, y, z) (16.64)
has a value at each pointr 1 = 1 (x, y, z); that is, a scalar (a number) is associated with each
point. Examples of scalar fields have been discussed in Chapter 10. A vectorfunction
of position, a vector field,
v 1 = 1 v(r) 1 = 1 v(x, y, z) (16.65)
T
l
=
d
dt
d
dt
r
=v
r
p
rp
r
11 zzz 11 11 p
d
dt
d
dt
d
dt
=−()
rF r
p
11 zz= 11
d
dt
F
p
t
=
d
d
.