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In the book "AP Physics 1 Essentials" by Dan Fullerton, question 6.32 entails a raft/boy system:

A 30kg raft with dimensions 3m X 3m that is motionless on a lake.A 40kg boy crosses from one corner of the raft to the other. Neglect friction.

Part A of the question asks how far the center of mass of the raft/boy system moves. The books states the answer as: "There is no interaction outside the system (no external force), therefore the center of mass will not move."

Shouldn't the center of mass shift from one corner to the other? Why does a system need an external force to shift its center of mass? Thanks for any help!

NoahGonzales
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3 Answers3

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The center of mass doesn't move from one part of the raft to the other, but the raft moves from one side of the center of mass to the other. The raft is not fixed, from the perspective of the lake. As the boy's feet push himself forward, they also push the raft backward.

Devsman
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Shouldn't the center of mass shift from one corner to the other?

Relative to the raft, the center of mass of the boy+raft system does move, just as you stated. Relative to the lake, the location of the center of mass of the boy+raft system does not move, just as the book stated.

What this means is that when the boy walks and changes the center of location of the center of mass relative to the raft, the raft moves in the opposite direction relative to the lake to keep the center of mass fixed.

Why does a system need an external force to shift its center of mass?

It's a consequence of conservation of momentum, or alternatively, of Newton's third law. Any force that the boy exerts on the raft is countered by an equal but opposite force that the raft exerts on the boy. Those internal forces cancel, and therefore play zero role in the motion of the center of mass relative to an inertial frame.

David Hammen
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What does the boy push against to propel him? --The raft. Newton's third law says that the forces on the boy and raft must be equal. So:

Newton's second law says Force is Mass time acceleration so the equal force accelerates the lighter raft more than the heavier boy. The same is true as the boy stops at the end and both accelerations are canceled.

How far does the raft move in what direction as the boy crosses it? --In the opposite direction from the way the boy walks, and a bit farther than he does.

Since center of mass is averages mass over distance, and the distance is changed proportional to mass the average does not change.

Momentum is conserved.

Zoom out until the boy and raft are a single object and it's course is set unless something acts on it according to Newtons first law.