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Chain with distance-dependent mass problem

  1. Aug 15, 2010 #1
    1. A particle of mass m is tied on one end of a very long chain which has a linear density μ (kg/m) and lies on a surface with the chain wound next to it. The particle is thrown upwards with an initial velocity V. Find the maximum height the particle is going to reach. My question is not what the exercise asks, it has more of a philosophical essence to it. You can read it in part 3.

    2. Relevant equations:
    We just take Newton's Second Law, F = dp/dt, so we have
    -(m+μx)g = d[(m+μx)υ]/dt
    and we solve the problem.

    3. Studying the particle's motion, my teacher said that while the particle is going upwards (υ>0), we consider m+μx as its mass. BUT when it is going downwards, that is when it starts falling, (υ<0) we consider m as its mass. I find this reasonable by instict (I mean when I visualize the phenomenon in my head) because the chain is supported by the surface/ground when the particle falls. Even my teacher told me that it is because of a normal force N applied to the chain by the ground thus neutralizing the weight of the chain. But my question is isn't the normal force N applied to the chain by the surface in the upwards motion, too? I don't see the differences in the forces that can convince me to say the downwards motion is different than the upwards one. I'd like someone to draw all the forces appearing in this problem in a sketch, so I can understand the phenomenon.

    Thanks in advance.

    Here's a sketch. The circle is the particle and the line which ends up in a spiral is the chain.
    Last edited: Aug 15, 2010
  2. jcsd
  3. Aug 17, 2010 #2

    Please, I just need a sketch of the forces applied to the particle and the chain. :'(
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