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Spring and pulley task

  1. Aug 5, 2017 #1
    1. The problem statement, all variables and given/known data
    2 bricks of mass m are located near the edge of a smooth horizontal surface, they are connected by an ideal weightless ,unstretched spring with a length of l0 and spring constant k. The brick which is closer to the edge is connected to another brick of the same mass by an ideal unstretchable rope going through a pulley. The lowest brick is held so that the rope above it hangs vertically.
    The lowest brick is released
    What's the minimal time τ after which the elongation of the spring ΔL will be maximal. Find ΔL.

    20645861_1928917583989680_1368339303_n.png 20631554_1928917840656321_1206854442_n.png
    2. Relevant equations


    3. The attempt at a solution
    Here's the solution until the point where I'm stuck. Sorry for the Russian text, it does not contain crucial information, please ignore it
    20632389_1928930010655104_1002110814_n.png
    a2-a1 is the acceleration of one end of the spring from the frame of reference of another end.
    Then these equations are given (y is vertical coordinate)
    20631623_1928927060655399_1063382409_n.png
    I understand why w2 is 3k/2m but why is the amplitude(A0) mg/3k? Can someone please explain? I only need an explanation for A0
    The next equations are
    20631814_1928927260655379_432063850_n.png
    20632625_1928928830655222_42064552_n.png
    20631703_1928928893988549_190674348_n.png
    At the beginning the system is in equilibrium and then x(0)=0, that leads to B being equal to 0 and and A0+A=0 because the spring isn't stretched at the beginning.
    20632455_1928929140655191_236803601_n.png
    20643868_1928929220655183_226754705_n.png
    20632337_1928929310655174_1726256519_n.png
    Thanks.
     
    Last edited: Aug 5, 2017
  2. jcsd
  3. Aug 5, 2017 #2

    haruspex

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    I have not tried to follow your equations, but I can suggest an easier way via a sequence of simplifications.
    First, model it as all happening on a horizontal surface, with a horizontal force mg applied to the rightmost block.
    Next, we can merge the two blocks connected by the inextensible string into a single block mass 2m.
    Clearly the average acceleration is g/3 to the right, so take a noninertial frame accelerating that way. In this frame, the left-hand block has a pseudoforce ..... to the left, and the 2m block has a pseudoforce ..... to the left, leaving it with a net force ..... to the right. (Fill in the blanks.)
    Finally, consider the common mass centre of the two blocks. How does this move in the reference frame? How do the two blocks move in relation to it?
     
  4. Aug 5, 2017 #3
    Thanks, your explanation hinted me why A0 is mg/3k. A0=amax/w2. And maximal acceleration amax after infinitesimal moment of time since the rightmost block has been released is gm/2m=g/2 because at that moment the spring is not deformed, so F=kΔx=k0=0, so we don't have to consider the spring and the leftmost block at that moment. So A0 =g*2m/(2*3*k)=mg/3k. Am I correct?
     
  5. Aug 5, 2017 #4

    haruspex

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    Looks good.
     
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