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410 CHAPTER 13 Dynamics of a Rigid Body

CONSERVATION OF ANGULAR
PROBLEM-SOLVING TECHNIQUES
MOMENTUM

The use of conservation of angular momentum in a problem 2 Then write an expression for the angular momentum at
involving rotational motion involves the familiar three steps another instant [Eq. (13.31)].
we used with conservation of momentum or of energy in
3 And then rely on conservation of angular momentum to
translational motion:
equate the two expressions [Eq. (13.32)].This yields one
1 First write an expression for the angular momentum at equation, which can be solved for an unknown quantity,
one instant of the motion [Eq. (13.30)]. such as the final angular speed.

✔ Checkup 13.3

QUESTION 1: A hoop and a uniform disk have equal radii and equal masses. Both are
spinning with equal angular speeds. Which has the larger angular momentum? By
what factor?
QUESTION 2: Two automobiles of equal masses are traveling around a traffic circle
side by side, with equal angular velocities. Which has the larger angular momentum?
QUESTION 3: You sit on a spinning stool with your legs tucked under the seat.
You then stretch your legs outward. How does your angular velocity change?
QUESTION 4: Consider the spinning skater described in Fig. 13.10. While
she brings her arms close to her body, does the rotational kinetic energy remain
constant?
QUESTION 5: Three children sit on a tire swing (see Fig. 13.14), leaning back-
ward as the wheel rotates about a vertical axis. What happens to the rotational
frequency if the children sit up straight?
(A) Frequency increases (B) Frequency decreases
(C) Frequency remains constant
FIGURE 13.14 Children on a spinning
tire swing.

13.4 TORQUE AND ANGULAR
MOMENTUM AS VECTORS

The rotational motion of a rigid body about a fixed axis is analogous to one-dimensional
translational motion. More generally, if the axis of rotation is not fixed but changes in
direction, the motion becomes three-dimensional. A wobbling, spinning top provides
an example of such a three-dimensional rotational motion. In this case, the torque and
the angular momentum must be treated as vectors, analogous to the force vector and
the momentum vector.The definitions of the torque vector and the angular-momen-
tum vector involve the vector cross product that we introduced in Section 3.4. When
a force F acts at some point with position vector r, the resulting torque vector is the
cross product of the position vector and the force vector:

torque vector r F (13.35)