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A Classical Mechanics challenge for fun

  1. Mar 14, 2017 #1
    I composed a problem and propose it here. I know the solution so it just for fun of the participants.

    There is a cylindrical bobbin of radius ##r##; the bobbin rotates about its central axis with angular velocity ##\omega=const>0##. An inextensible weightless string is coiled around the bobbin. The string is very thin and very long. On a free end of the string there is a point mass ##m##. Let ##l=l(t)## stand for the length of the free part of the string. There is no gravity.
    Initially the value ##l_0=l(0)>0## and the velocity ##v=\frac{d}{dt} l\mid_{t=0}## are given. Moreover it is known that ##-\omega r<v<0##. Find minimal value of the function ##l(t)## that is ##\min_{t\ge 0}l(t)## -- ?.
    We consider motion such that the string remains strained for all time

    506af59672cf.png
     
    Last edited by a moderator: Mar 15, 2017
  2. jcsd
  3. Mar 14, 2017 #2
    Sounds like an interesting problem. I've seen similar ones in graduate qualifiers in the classical mechanics section.
     
  4. Mar 16, 2017 #3
    After some discussion we are allowing this as a stand alone but unofficial challenge. Good luck!
     
  5. Mar 17, 2017 #4
    the Lagrangian is
    $$L\Big(l,\frac{d l}{dt}\Big)=\frac{m}{2}\Big(l^2\omega^2+\frac{l^2}{r^2}\Big(\frac{d l}{dt}\Big)^2 \Big);$$
    the answer is $$l_0\sqrt{1-\Big(\frac{v}{\omega r}\Big)^2}$$
     
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