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412 CHAPTER 13 Dynamics of a Rigid Body

Suppose that the rod of the dumbbell described in the
z EXAMPLE 11
preceding example is welded to an axle inclined at an angle
with respect to the rod.The dumbbell rotates with angular velocity about this
axis, which is supported by fixed bearings (see Fig. 13.18). Find the angular momen-
Angular
momentum tum about an origin on the axis, at the center of mass.
L r p need
not lie along axis 90° – SOLUTION: Each particle executes circular motion, but since the origin is not at
of rotation.
the center of the circle, the angular momentum is not the same as in Example 10.
The distance between each particle and the axis of rotation is
L 1
m R r sin
R

and the magnitude of the velocity of each particle is
r r 1
O O
y v R r sin
L 2 r r 2 2 2
The direction of the velocity is perpendicular to the position vector. Hence the
R angular-momentum vector of each mass has a magnitude
m
2
ƒ L ƒ ƒ L ƒ m ƒ r v ƒ mrv m r sin (13.39)
1 2
The direction of the angular-momentum vector of each mass is perpendicular
FIGURE 13.18 A rotating dumbbell ori- to both the velocity and the position vectors, as specified by the right-hand rule.
ented at an angle with the axis of rotation. The angular-momentum vector of each mass is shown in Fig. 13.18; these vectors
are parallel to each other, they are in the plane of the axis and the rod, and they
make an angle of 90 with the axis.The total angular momentum is the vector
sum of these individual angular momenta. This vector is in the same direction as
the individual angular-momentum vectors, and it has a magnitude twice as large
as either of those in Eq. (13.39):
2
L 2m r sin (13.40)
As the body rotates, so does the angular-momentum vector, remaining in the
plane of the axis and the rod. If at one instant the angular momentum lies in the
z–y plane, a quarter of a cycle later it will lie in the z–x plane, etc.
COMMENT: Note that the z component of the angular momentum is
2
2
2
L L cos(90 ) 2m r sin cos(90 ) 2m r sin
z
This can also be written as
2
L 2m
R (13.41)
z
where R r sin is the perpendicular distance between each mass and the axis
2
of rotation. Since 2mR is simply the moment of inertia of the two particles about
the z axis, Eq. (13.41) is the same as

L I (13.42)
z
As we will see below, this formula is of general validity for rotation around a
fixed axis.

The preceding example shows that the angular-momentum vector of a rotating body
need not always lie along the axis of rotation. However, if the body is symmetric about
the axis of rotation, then the angular-momentum vector will lie along this axis. In such
a symmetric body, each particle on one side of the axis has a counterpart on the other