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Plank with block of mass on top executing vertical SHM

  1. Jan 30, 2015 #1
    1. The problem statement, all variables and given/known data
    A plank on which a mass of mass m is put, is executing vertical SHM according to the equation y = sinωt + √3cosωt. At what time does the mass break of the plank and what is the value for the mass to break off ?

    2. Relevant equations
    The modified equation of motion looks like: y = 2sin(ωt + π/3)
    Maximum chance of break off is at extreme position. So mg - N = mω2A

    3. The attempt at a solution.
    My doubt is that why does the maximum chance for the mass to break off occur at an extreme position of the plank and that too at the positive extreme (A) and not the negative extreme (-A)? Can it be explained quantitatively rather than qualitatively?
     
  2. jcsd
  3. Jan 30, 2015 #2

    DEvens

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    What has to happen for the block to leave the plank? What forces are involved?

    I did not check the modified equation of motion, so you should do that if you have not already.

    What is the N in mg - N = mω2A ?

    Who says the block leaves at the maximum position?
     
  4. Jan 30, 2015 #3
    The question asks the time when the block breaks off and the minimum ω needed to do so.
    N is the normal reaction acting on the block because of the contact between block and plank.
     
  5. Jan 30, 2015 #4

    haruspex

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    Put that information aside for the moment. At what point in the cycle do you think there's the greatest chance of the mass leaving the plank?
     
  6. Jan 30, 2015 #5
    At the topmost point of course. That's what my logic and intuition says. But I am not able to find a suitable quantitative explanation for this.
     
  7. Jan 30, 2015 #6

    haruspex

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    Think about forces. What does "stays on the plank" mean in terms of forces?
     
  8. Jan 30, 2015 #7
    Well, the block stays on the plank but its not glued to it or anything like that. So, it can break off if such a situation occurs, can't it?
     
  9. Jan 30, 2015 #8

    haruspex

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    Exactly, it's not glued on. What are the forces on it? What will be the relationship between those forces when it separates from the plank?
     
  10. Jan 31, 2015 #9
    When it breaks off then the normal reaction between the block and the plank is 0. The forces acting on the block are represented by the following equation
    mg - N = mω2A.

    But my question is that why is the net acceleration of the block mω2A? Isn't that supposed to be towards the mean position and if it is then how would the block ever break off if its acceleration is towards the mean position, i.e vertically down at the topmost point and vertically upwards at the bottom-most point?
     
  11. Jan 31, 2015 #10

    haruspex

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    In general, it is a function of time.
    Because to become detached, the block only needs to accelerate downwards more slowly than the plank.
     
  12. Jan 31, 2015 #11
    But both the plank and the block have an acceleration of ω2A downwards at the topmost position. Then how can one be slower than the other?
     
  13. Jan 31, 2015 #12

    haruspex

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    As long as the block has that acceleration downwards it will stay with the plank. But how is it guaranteed to have that acceleration down? Its acceleration is a consequence of the forces on it, not of a wish to stay in SHM.
     
  14. Jan 31, 2015 #13
    So
    So the forces acting on the block would be its own weight and the normal reaction then. But how does that determine whether the acceleration of the block would be lesser than the plank or not?
     
  15. Jan 31, 2015 #14

    haruspex

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    The plank will follow SHM, so you know the acceleration of that. When will the forces acting on the block fall short of what is required? Consider what the maximum downward acceleration of the block can be.
     
  16. Jan 31, 2015 #15
    The plank follows SHM, and so at all instants when the plank is moving up, it has acceleration towards the mean position and since the block is sitting on top of it, it too has an acceleration towards the mean position even though it is going up. This case is a bit confusing. Because I compare it with the case of an elevator which is accelerating upwards and a man is standing inside it on a weighing scale. As the elevator accelerates upwards, the weighing scale shows the weight of the man more than what it actually is. This is because the normal reaction faced by the man is more than his weight downward (and hence contributing to an acceleration equal to that of the elevator). But this happens when the elevator (in this case a plank) is accelerating upwards. But the plank is accelerating downwards even though it is going up and so the normal reaction experienced by the block is less than its weight downwards and so it has no reason to break off even at the top most point.

    What I am trying to come at is that even though the whole system (plank + block) are going up, their net acceleration is down. But I am not able to figure out the case when the accelerations of the two are both downwards but unequal.
     
  17. Jan 31, 2015 #16

    haruspex

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    Answer this: what is the maximum downward acceleration the block can ever have (given that it is not glued on).
     
  18. Jan 31, 2015 #17
    2A + mg - N?
     
  19. Jan 31, 2015 #18

    haruspex

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    No. You're overthinking it. There are only two forces acting on the block. The downward force is constant. The upward force has a minimum.
     
  20. Jan 31, 2015 #19
    mg - N?
     
  21. Jan 31, 2015 #20

    haruspex

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    Ok, but the maximum value of that is?
     
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