9.6

Angular Momentum

In Chapter 7 the linear momentum p of an object is defined as the product of its mass m and linear velocity v; that is, p = mv. For rotational motion the analogous concept is called the angular momentum L. The mathematical form of angular momentum

| Definition of Angular Momentum
| The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia I and its angular velocity with respect to that axis:

| (9.10)

Requirement: must be expressed in rad/s.
SI Unit of Angular Momentum: kg · m2/s

is analogous to that of linear momentum, with the mass m and the linear velocity v being replaced with their rotational counterparts, the moment of inertia I and the angular velocity .

CONCEPTS AT A GLANCE Linear momentum is an important concept in physics because the total linear momentum of a system is conserved when the sum of the average external forces acting on the system is zero. Then, the final total linear momentum Pf and the initial total linear momentum P0 are the same: Pf = P0 Figure 7-8 outlines the conceptual development of the conservation of linear momentum. The Concepts-at-a-Glance chart in Figure 9-26 outlines a similar development for angular momentum. In constructing this chart, we recall that each translational variable is analogous to a corresponding rotational variable, as Table 9-2 indicates. Therefore, if we replace “force” by “torque” and “linear momentum” by “angular momentum” in Figure 7-8, we obtain Figure 9-26.

Figure 9-26 CONCEPTS AT A GLANCE The impulse–momentum theorem for rotational motion leads to the principle of conservation of angular momentum when the sum of the external torques is zero. This concept diagram is similar to that in Figure 7.8, which shows how the conservation of linear momentum arises from the impulse–momentum theorem. The angular momentum of each of these spinning skaters is conserved while they are in the air, assuming that air resistance is negligible. (Paolo Cocco/AFP/Getty Images Sport Services)

| Principle of Conservation of Angular Momentum
| The total angular momentum of a system remains constant (is conserved) if the net average external torque acting on the system is zero.

This chart indicates that when the sum of the average external torques is zero, the final and initial angular momenta are the same: Lf = L0 which is the principle of conservation of angular momentum.
Example 14 illustrates an interesting consequence of the conservation of angular momentum.

| Example 14 | | | A Spinning Skater

In Figure 9-27a an ice skater is spinning with both arms and a leg outstretched. In Figure 9-27b she pulls her arms and leg inward. As a result of this maneuver, her spinning motion changes dramatically. Using the principle of conservation of angular momentum, explain how and why it changes.

Figure 9-27 (a) A skater spins slowly on one skate, with both arms and one leg outstretched. (b) As she pulls her arms and leg in toward the rotational axis, her moment of inertia I decreases, and her angular velocity increases.

Reasoning and Solution Choosing the skater as the system, we can apply the conservation principle provided that the net external torque produced by air resistance and by friction between the skates and the ice is negligibly small. We assume that it is. Then the skater in Figure 9-27a would spin forever at the same angular velocity, since her angular momentum is conserved in the absence of a net external torque. In Figure 9-27b the inward movement of her arms and leg involves internal, not external, torques and, therefore, does not change her angular momentum. But angular momentum is the product of the moment of inertia I and angular velocity . By moving the mass of her arms and leg inward, the skater decreases the distance r of the mass from the axis of rotation and, consequently, decreases her moment of inertia I (). If the product of I and is to remain constant, then must increase. Thus, the consequence of pulling her arms and leg inward is that she spins with a larger angular velocity.
Related Homework: Conceptual Questions 20, 21, 22 and Problem 56

The next example involves a satellite and illustrates another application of the principle of conservation of angular momentum.

The physics of a spinning ice skater. The physics of a satellite in orbit about the earth.

| Example 15 | | | A Satellite in an Elliptical Orbit

An artificial satellite is placed into an elliptical orbit about the earth, as illustrated in Figure 9-28. Telemetry data indicate that its point of closest approach (called the perigee) is from the center of the earth, and its point of greatest distance (called the apogee) is from the center of the earth. The speed of the satellite at the perigee is . Find its speed vA at the apogee.

Figure 9-28 A satellite is moving in an elliptical orbit about the earth. The gravitational force exerts no torque on the satellite, so the angular momentum of the satellite is conserved.

Reasoning The only force of any significance that acts on the satellite is the gravitational force of the earth. However, at any instant, this force is directed toward the center of the earth and passes through the axis about which the satellite instantaneously rotates. Therefore, the gravitational force exerts no torque on the satellite (the lever arm is zero). Consequently, the angular momentum of the satellite remains constant at all times.
Solution Since the angular momentum is the same at the apogee (A) and the perigee (P), it follows that . Furthermore, the orbiting satellite can be considered a point mass, so its moment of inertia is (see Equation 9.4). In addition, the angular speed of the satellite is related to its tangential speed vT by (Equation 8.9). If these relations are used at the apogee and perigee, the conservation of angular momentum gives the following result:

The answer is independent of the mass of the satellite. The satellite behaves just like the skater in Figure 9-27, because its speed is greater at the perigee, where the moment of inertia is smaller.

The result in Example 15 indicates that a satellite does not have a constant speed in an elliptical orbit. The speed changes from a maximum at the perigee to a minimum at the apogee; the closer the satellite comes to the earth, the faster it travels. Planets moving around the sun in elliptical orbits exhibit the same kind of behavior, and Johannes Kepler (1571–1630) formulated his famous second law based on observations of such characteristics of planetary motion. Kepler’s second law states that, in a given amount of time, a line joining any planet to the sun sweeps out the same amount of area no matter where the planet is on its elliptical orbit, as Figure 9-29 illustrates. The conservation of angular momentum can be used to show why the law is valid, by means of a calculation similar to that in Example 15.

Figure 9-29 Kepler’s second law of planetary motion states that a line joining a planet to the sun sweeps out equal areas in equal time intervals.

Check your knowledge of the main ideas presented in Sections 9.4–9.6:

· | Newton’s Second Law for Rotational Motion
· | Rotational Work and Energy
· | Angular Momentum

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