## Angular Momentum

11.2 Angular Momentum

### Learning Objectives

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

- Describe the vector nature of angular momentum
- Find the total angular momentum and torque about a designated origin of a system of particles
- Calculate the angular momentum of a rigid body rotating about a fixed axis
- Calculate the torque on a rigid body rotating about a fixed axis
- Use conservation of angular momentum in the analysis of objects that change their rotation rate

Why does Earth keep on spinning? What started it spinning to begin with? Why doesn’t Earth’s gravitational attraction not bring the Moon crashing in toward Earth? 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?

### Angular Momentum of a Single Particle

**Practice!**

Practice 11.2.1 |
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Imagine a rigid pole sticking up through the ice in a frozen pond. A skater glides rapidly toward the pole, aiming a little to the side so she doesn’t hit it. As she passes the pole she reaches out to her side and grabs it, an action that causes her to move in a circular path around the pole. Let her mass be m. What would be her angular momentum relative to the pole at the instant she is a distance d from the pole if she were skating at a speed v along a straight path that is a perpendicular distance a from the pole? |

(a) zero |

(b) mvd |

(c) mva |

### Angular Momentum of a Rigid Body

**Practice!**

Practice 11.2.2 |
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A solid sphere and a hollow sphere have the same mass and radius. They are rotating with the same angular speed. Which one has more angular momentum? |

(a) the solid sphere |

(b) the hollow sphere |

(c) both have the same angular momentum |

(d) impossible to determine |

Practice 11.2.3 |
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Three identical wheels are all spinning with the same angular velocity. The total angular momentum of the 3-wheel system has magnitude L. One of the three wheels is flipped upside-down, while the magnitude of its angular velocity remains constant. The new angular momentum of the 3-wheel system has magnitude… |

(a) L (the same as before) |

(b) (2/3)L |

(c) (1/3)L |

(d) some other value |

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

A ball is attached to one end of a piece of string. You hold the other end of the string and whirl the ball in a circle around your hand.

(a) If the ball moves at a constant speed, is its linear momentum constant? Why or why not?

(b) Is its angular momentum constant? Why or why not?