10.5 Angular momentum and its conservation

Learning objectives

By the end of this section, you will be able to:

• Understand the analogy between angular momentum and linear momentum.
• Observe the relationship between torque and angular momentum.
• Apply the law of conservation of angular momentum.

The information presented in this section supports the following AP® learning objectives and science practices:

• 4.D.2.1 The student is able to describe a model of a rotational system and use that model to analyze a situation in which angular momentum changes due to interaction with other objects or systems. (S.P. 1.2, 1.4)
• 4.D.2.2 The student is able to plan a data collection and analysis strategy to determine the change in angular momentum of a system and relate it to interactions with other objects and systems. (S.P. 2.2)
• 4.D.3.1 The student is able to use appropriate mathematical routines to calculate values for initial or final angular momentum, or change in angular momentum of a system, or average torque or time during which the torque is exerted in analyzing a situation involving torque and angular momentum. (S.P. 2.2)
• 4.D.3.2 The student is able to plan a data collection strategy designed to test the relationship between the change in angular momentum of a system and the product of the average torque applied to the system and the time interval during which the torque is exerted. (S.P. 4.1, 4.2)
• 5.E.1.1 The student is able to make qualitative predictions about the angular momentum of a system for a situation in which there is no net external torque. (S.P. 6.4, 7.2)
• 5.E.1.2 The student is able to make calculations of quantities related to the angular momentum of a system when the net external torque on the system is zero. (S.P. 2.1, 2.2)
• 5.E.2.1 The student is able to describe or calculate the angular momentum and rotational inertia of a system in terms of the locations and velocities of objects that make up the system. Students are expected to do qualitative reasoning with compound objects. Students are expected to do calculations with a fixed set of extended objects and point masses. (S.P. 2.2)

Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum.

By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum     $L$ as

This equation is an analog to the definition of linear momentum as $p=\text{mv}$ . Units for linear momentum are $\text{kg}\cdot \text{m}\text{/s}$ while units for angular momentum are $\text{kg}\cdot {\text{m}}^2\text{/s}$ . As we would expect, an object that has a large moment of inertia $I$ , such as Earth, has a very large angular momentum. An object that has a large angular velocity $\omega$ , such as a centrifuge, also has a rather large angular momentum.