## Center of Mass

9.6 Center of Mass

### Learning Objectives

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

- Explain the meaning and usefulness of the concept of center of mass
- Calculate the center of mass of a given system
- Apply the center of mass concept in two and three dimensions
- Calculate the velocity and acceleration of the center of mass

**Discuss!**

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

What is the center-of-mass position vector for the system of masses given in the figure? Express your answer in terms of m and d.

### Motion of the Center of Mass

**Practice!**

Practice 9.6.1 |
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At the instant a 2.0-kg particle has a velocity of 4.0 m/s in the positive x direction, a 3.0-kg particle has a velocity of 5.0 m/s in the positive y direction. What is the speed of the center of mass of the two-particle system? |

(a) 3.8 m/s |

(b) 3.4 m/s |

(c) 5.0 m/s |

(d) 4.4 m/s |

(e) 4.6 m/s |

Practice 9.6.2 |
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Block A is initially moving to the right at 6.00 m/s, while block B is initially at rest. The surface they move on is level and frictionless. What is the velocity of the center of mass of the two blocks before the blocks collide? |

(a) 6.00 m/s, to the right |

(b) 3.00 m/s, to the right |

(c) 1.50 m/s, to the right |

(d) zero |

(e) Not enough information is given to decide. |

Practice 9.6.3 |
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Block A is initially moving to the right at 6.00 m/s, while block B is initially at rest. The surface they move on is level and frictionless. What is the velocity of the center of mass of the two blocks after the blocks collide? |

(a) 6.00 m/s, to the right |

(b) 3.00 m/s, to the right |

(c) 1.50 m/s, to the right |

(d) zero |

(e) Not enough information is given to decide. |

### Momentum of the Center of Mass

**Discuss!**

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

Two astronauts (one with mass = m, the other with mass = 1.5m) at rest in outer space are connected by a light rope. They begin to pull towards each other. Where do they meet?

**Practice!**

Practice 9.6.4 |
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You are sitting at the left end of a canoe in the middle of a lake. You carefully begin to walk towards the right end of the canoe. (Neglect friction between water and canoe) As viewed from the edge of the lake, as you walk, the CM of the person-canoe system… |

(a) moves to the right |

(b) moves to the left |

(c) stays in the same spot |

(d) It depends on how fast you move. |

Practice 9.6.5 |
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A man weighs exactly as much as his 20 foot long canoe. Initially he stands in the center of the motionless canoe, a distance of 20 feet from shore. Next he walks toward the shore until he gets to the end of the canoe. What is his new distance from the shore? (There is no horizontal force on the canoe by the water). |

(a) 20 ft |

(b) 15 ft |

(c) 10 ft |

(d) 5 ft |

(e) 0 ft |