Calculating Centre of Mass on a Board on Ice

AI Thread Summary
The discussion centers on calculating the center of mass for a system consisting of a person and a board on a frictionless ice surface. The initial calculation for the center of mass is straightforward, yielding (ml)/(2(M+m)). However, there is confusion regarding the distance the person moves relative to their starting position, with differing answers presented. One participant argues that the movement should account for the board's motion in the opposite direction, suggesting a different formula. Ultimately, the consensus emphasizes that due to the frictionless nature of the surface, the board and person must move in a way that maintains the system's center of mass.
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Board on Ice

1. A person of mass M is standing at one end of a board of mass m and length l. The board rests upon a frictionless ice surface and it's mass is uniformly distributed along its length. Calculate how far from the person, d, the centre of mass is of the system. The person then walks to the centre of the board and stops (assuming friction is enough). How far from his starting position (relative to the ice surface) has he moved?



2. centre of mass = 1/M\Sigmamr



The Attempt at a Solution



All right, the centre of mass is easy to calculate and it's just (ml)/(2(M+m) and I didn't have problems with that. It's the next part which I am unsure of the correct answer. My friends have got an answer of (ml)/(2(M+m) as his total distance moved but I get a slightly different answer of (l/2)(1 - (m)/(M+m)).

My reasoning is that it has to be slightly less then (l/2) as he walks that distance on the board but the board has moved in the opposite direction so the distance from the original starting point is slightly less.

The answer of (ml)/(2(M+m)) seems obvious and I'm not 100% it's that or am I just over thinking a simple question?


Thanks.
 
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The answer of (ml)/(2(M+m)) seems obvious

Your friends are wrong. As the surface is frictionless, there are no horizontal forces acting on the (man+board) system. Hence, as the person's CM moves to the left (say he was standing on the right end of the board), the board must move to the right. Hence, the person and the board "meet somewhere in the middle". Sort of a conveyor belt.

See if this new insight helps you with your solution.

--------
Assaf
http://www.physicallyincorrect.com/"
 
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ozymandias said:
Your friends are wrong. As the surface is frictionless, there are no horizontal forces acting on the (man+board) system. Hence, as the person's CM moves to the left (say he was standing on the right end of the board), the board must move to the right. Hence, the person and the board "meet somewhere in the middle". Sort of a conveyor belt.

See if this new insight helps you with your solution.

--------
Assaf
http://www.physicallyincorrect.com/"


Yah, that's exactly the reasoning I used to get an answer of (l/2)(1 - (m)/(2(M+m)). I'll try to recheck my answer again to see if I've made any mistake. Thanks for the help. If anyone else has any suggestions I'd love to hear them.
 
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If you will detail your solution I could point out any mistakes I see.

--------
Assaf
http://www.physicallyincorrect.com/"
 
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