## Moment of Inertia and Rotational Kinetic Energy

10.4 Moment of Inertia and Rotational Kinetic Energy

### Learning Objectives

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

- Describe the differences between rotational and translational kinetic energy
- Define the physical concept of moment of inertia in terms of the mass distribution from the rotational axis
- Explain how the moment of inertia of rigid bodies affects their rotational kinetic energy
- Use conservation of mechanical energy to analyze systems undergoing both rotation and translation
- Calculate the angular velocity of a rotating system when there are energy losses due to nonconservative forces

### Rotational Kinetic Energy and Moment of Inertia

**Practice!**

Practice 10.4.1 |
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The figure shows a system of four particles joined by light, rigid rods. Assume a = b and M is larger than m. About which of the coordinate axes does the system have the smallest moment of inertia? |

(a) the x axis |

(b) the y axis |

(c) the z axis |

(d) The moment of inertia has the same small value for two axes. |

(e) The moment of inertia is the same for all axes. |

Practice 10.4.2 |
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Consider a rod of uniform density with an axis of rotation through its center and an identical rod with the axis of rotation through one end. Which has the larger moment of inertia? |

(a) C: the one rotating about its center |

(b) E: the one rotating about its end |

(c) They have the same moment of inertia. |

Pause & Predict 10.4.1 |
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What is the moment of inertia of this system of particles when the system rotates about an axis of rotation through the center of mass as shown? |

(a) 2.82 kg•m^{2} |

(b) 5.63 kg•m^{2} |

(c) 11.2 kg•m^{2} |

(d) 45.2 kg•m^{2} |

(e) 22.6 kg•m^{2} |

Pause & Predict 10.4.2 |
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This object rotates with an angular speed of 32 rad/s. What is its rotational kinetic energy? |

(a) 7.3 × 10^{4} J |

(b) 3.6 × 10^{3} J |

(c) 4.6 × 10^{3} J |

(d) 6.9 × 10^{2} J |

(e) 2.2 × 10^{4} J |

**Practice!**

Practice 10.4.3 |
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Max (15 kg) and Maya (12 kg) are riding on a merry-go-round that rotates at a constant speed. Max is sitting on the edge of the merry-go-round, 2.4 m from the center, and Maya is 1.2 m from the center. Considering Max and Maya to be one system of masses, what is their moment of inertia measured with respect to the center of the merry-go-round? |

(a) 50 kg•m^{2} |

(b) 104 kg•m^{2} |

(c) 86 kg•m^{2} |

(d) 17 kg•m^{2} |

(e) 1500 kg•m^{2} |

Practice 10.4.4 |
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Max (15 kg) and Maya (12 kg) are riding on a merry-go-round that rotates at a constant speed. Max is sitting on the edge of the merry-go-round, 2.4 m from the center, and Maya is 1.2 m from the center. The merry-go-round is a solid disk with a radius of 2.4 m and a mass of 230 kg. What is the moment of inertia of Max, Maya, and the merry-go-round? |

(a) 766 kg•m^{2} |

(b) 1430 kg•m^{2} |

(c) 214 kg•m^{2} |

(d) 662 kg•m^{2} |

(e) 104 kg•m^{2} |

Practice 10.4.5 |
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Max (15 kg) and Maya (12 kg) are riding on a merry-go-round that rotates at a constant speed. Max is sitting on the edge of the merry-go-round, 2.4 m from the center, and Maya is sitting somewhere between the edge and the center. The merry-go-round is a solid disk with a radius of 2.4 m, a mass of 230 kg, and is rotating at a constant rate of 0.75 rev/s. The system, which includes Max, Maya, and the merry-go-round, has 8700 J of rotational kinetic energy. How far away is Maya from the center of the merry-go-round? |

(a) 50 m |

(b) 2.9 m |

(c) 1.7 m |

(d) 1.5 m |

(e) 1.9 m |