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A person of mass $m$ is standing on one end of a frictionless plank of mass $M$ and length $L$ and floating in the water. The person moves from one end to another and stops. The displacement of the plank is

Soltn: Take origin to be left end of the rod, $$x_{com} = \frac{mL + M \frac{L}{2} }{m+M},$$ in the final state, the planck's com moves by $\Delta x$ and the man would be a total of $\Delta x$ from the original origin. Hence, $$x_{com} = \frac{m \Delta x + M( \frac{L}{2} + \Delta x)}{m+M},$$ equating the two expressions from coms and cancelling, $\Delta x = \frac{mL}{M+m}$.

Going by the formulas, this question is simple to do. Calculate the COM in the beginning, and calculate the com in the end state, and note they are the same point. But, is it possible to show how the motion will play out mechanistically?

I can't seem to understand why the plank moves because since there is no friction, there will be no forces between the plank and man. I know due to lack of friction the man will slide into the edge of the rod, but still, the rod moving conclusion doesn't make sense.

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    $\begingroup$ Doesn't make sense to me either unless he moved along the plank by pulling himself using the end of the plank rather than trying to walk or pull using the edges which wouldn't work because frictionless. Or he could blow air or paddle the water but that wouldn't affect the plank. $\endgroup$
    – DKNguyen
    Commented Aug 12, 2021 at 13:50
  • $\begingroup$ I am not sure, but I think the language is a bit vague. I believe the friction is there between man and plank, but absent between plank and water. So if you take man+plank as system then there is no external force and you can conserve the COM position. @Buraian does it make sense? $\endgroup$
    – sslucifer
    Commented Aug 12, 2021 at 13:56
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    $\begingroup$ Hmm... but I can't understand what force that person exerts or is exerted on to accelerate when sliding. $\endgroup$
    – ACB
    Commented Aug 12, 2021 at 14:22
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    $\begingroup$ Ah I think thats the answer, he can't move without pushing against something? So he pushes the boat when he moves and then the displacement thing $\endgroup$
    – Brian
    Commented Aug 12, 2021 at 14:31
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    $\begingroup$ Do you mean, how can we "walk" on a wet floor? We walk carefully. We can do that at all because it is not completely frictionless. How do we "slip" on it? We can do that because it's close to frictionless. $\endgroup$
    – Solomon Slow
    Commented Aug 12, 2021 at 15:51

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Unfortunately, the question is poorly phrased. As emphasized in comments, there must be frictional forces acting between the man and the plank. Let's look at the question again. It says that it is a frictionless plank. Let's assume however man is initially sliding on the plank with the help of an external force gained before landing on the plank. But at the same time the question says,

The person moves from one end to another and stops

Is this possible? No. This is similar to a car braking on a frictionless surface. You can imagine what happens. It can't stop.

Hence your question cannot be solved unless it was intended to assume there is no friction between water and the plank, not man and the plank.

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  • $\begingroup$ I have concluded that the center of mass calculation, while correct, is a change of 'state' which can't not physically happen in the circumstance $\endgroup$
    – Brian
    Commented Aug 12, 2021 at 16:22
  • $\begingroup$ Though no use, unless something is physically possible... And unless your question means what I suggested at the end of my answer, you can assume there is no plank at all. The man will slide on and fall into the water at the other end. But the plank won't move. So your calculation is not correct. (But question is worse than your solution) $\endgroup$
    – ACB
    Commented Aug 12, 2021 at 16:34
  • $\begingroup$ If we consider the plank as frictionless, the man can't stop himself, and there will be no need of centre of mass being fixed as well. Because there is no interaction between the man and the plank. Like in an ice hockey game, one player slides and another one is looking at him. No forces acting between them. There will be no need of that centre of mass calculation. Because we define the momentum conservation of com for a system, and to consider as a system there should be interactions between parts. $\endgroup$
    – ACB
    Commented Aug 13, 2021 at 4:53
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Friction can't be absent between the man and the plank otherwise the man will not be able to move I think the question would be framed like friction is absent between water and plank

Now considering there is friction between man and plank As the man tries to move forward the plank applies friction force in the forward direction and so the man can move forward and by Newton's third law, a backward friction force will be applied on the plank due to which the plank will move back

You can take the example of walking on a really slippery carpet to compare it

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In order for the man to accelerate with respect to the plank he needs to be able to push back on the plank with a force and the plank needs to push forward on the man with an equal and opposite force per Newton’s 3. That requires static friction between the man and the plank.

If we can ignore friction between the plank and the water, then there are no external horizontal forces acting on the plank man.combination. Given the man and plank are not initially moving relative to the shore, then the center of mass of the man plank combination will not move. (Conservation of momentum).

Hope this helps

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  • $\begingroup$ Hmm right but then what about the center of mass calculation that I have made? Are you saying that the calculation is not practically feasible. $\endgroup$
    – Brian
    Commented Aug 12, 2021 at 15:07
  • $\begingroup$ First, it doesn’t matter if there’s friction between the man and plank, the COM will not move as long as there’s no friction between the plank and the water. Regarding your COM calculation, it’s not clear to me exactly what $\Delta x$ is. With respect to the shore $\Delta x$ will be different for the man and the plank unless $m=M$. $\endgroup$
    – Bob D
    Commented Aug 12, 2021 at 17:08

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