## Calculating Moments of Inertia

10.5 Calculating Moments of Inertia

### Learning Objectives

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

- Calculate the moment of inertia for uniformly shaped, rigid bodies
- Apply the parallel axis theorem to find the moment of inertia about any axis parallel to one already known
- Calculate the moment of inertia for compound objects

### Rotational Inertia for Rigid Bodies

**Practice!**

Practice 10.5.1 |
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Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold. Which one has the biggest moment of inertia about an axis through its center? |

(a) Solid aluminum |

(b) Hollow gold |

(c) Same |

Practice 10.5.2 |
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The three objects shown here all have the same mass and the same outer radius. Each object is rotating about its axis of symmetry (shown in blue). All three objects have the same rotational kinetic energy. Which object is rotating fastest? |

(a) Object A is rotating fastest. |

(b) Object B is rotating fastest. |

(c) Object C is rotating fastest. |

(d) Two of these are tied for fastest. |

(e) All three rotate at the same speed. |

**Discuss!**

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

While working on your latest novel about settlers crossing the Great Plains in a wagon train, you get into an argument with your co-author regarding the moment of inertia of an actual wooden wagon wheel. The 70-kg wheel is 120 cm in diameter and has heavy spokes connecting the rim to the axle. Your co-author claims that you can approximate using the moment of inertia I=MR^{2} (like for a hoop) but you anticipate the moment of inertia will be significantly less than that because of the mass located in the spokes. To find the moment of inertia experimentally, you mount the wheel on a low-friction bearing and then wrap a light rope around the outside of the rim to which you attach a 20-kg bag of sand. When the bag is released from rest, it drops 3.77 m in 1.6 s during which time the wheel rotates through an angle of 2π radians. (Hint: use energy considerations.)

### Parallel-Axis Theorem

**Discuss!**

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

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass M and radius R about an axis perpendicular to the hoop’s plane at an edge.