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Motion of masses connected by a string over a pulley

  1. Feb 12, 2019 at 8:04 AM #1
    1. The problem statement, all variables and given/known data
    Two particles A and B of masses m and 2m are attached to the ends of light inextensible string which passes over smooth fixed pulley. The particles are released from rest and when particle B has moved h meters, it hits the ground and does not rebound.
    (i) Find the speed of the particles just before B hits the ground
    (ii) Find the direction and magnitude of the impulse that the particle B exerting on the ground
    (iii) In the subsequent motion, B will rest on the ground until it is jerked into motion again. Find the period of time when B is resting on the ground before it is first jerked into motion
    (iv) Find the speed when B first leaves the ground
    (v) After jerked into motion, B rises up and then drops to hit the ground again. Find the distance B travels between the first hit and the second hit on the ground
    (vi) The motion continues with B hitting the ground and jerked into motion indefinitely. Find the total distance travelled by B from the instance that the particles are released from rest.

    2. Relevant equations
    v2 = u2 + 2ad
    v = u + at
    d = u.t + 1/2 at2
    ΣF = m.a

    3. The attempt at a solution
    (i) acceleration of system = g/3

    v2 = u2 + 2ad
    [itex]= \sqrt {\frac{2gh}{3}} [/itex]

    (ii) Impulse on B = m . Δv = [itex] m . (0 - \sqrt {\frac{2gh}{3}}) [/itex]
    [itex] = m . \sqrt {\frac{2gh}{3}}[/itex]

    So the impulse on ground is downwards

    (iii) Period of time when B is resting = twice the period of time for A to travel (after B hits the ground) to max. height, so:
    [itex]v = u - gt[/itex]
    [itex] 0 = \sqrt {\frac{2gh}{3}} - gt [/itex]
    [itex] t = \sqrt {\frac{2h}{3g}} [/itex]

    (iv) Speed when B leaves the ground = speed when B almost hits the ground [itex]= \sqrt {\frac{2gh}{3}} [/itex]

    (v) distance of B = twice the distance when it accelerates upwards + decelerate to stop (when A hits the ground)

    Distance when B decelerates to stop:
    v2 = u2 - 2gd
    0 = 2gh/3 - 2gd
    d = h/3

    So total distance = 4h + 2(h/3) = 14h/3


    Are answer from (i) to (v) correct?


    For (vi), the answer is infinite?

    Thanks
     
  2. jcsd
  3. Feb 12, 2019 at 9:44 AM #2

    BvU

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    Hi,

    Don't forget the factor 2 in iii

    I don't understand your argument for iv : B does not rebound and the momentum from B is lost. What happens when A pulls the string taut again ?
     
  4. Feb 12, 2019 at 9:36 PM #3
    Hi BvU

    Yes, my bad.

    Yeah, the system should be decelerating instead of accelerating o:). The deceleration = g/3

    But I do not know how to find the speed of B when it leaves the ground. Won't it be the same as speed of A when the string becomes taut, which is [itex]\sqrt {\frac{2gh}{3}}[/itex]?

    Thanks
     
  5. Feb 12, 2019 at 9:54 PM #4

    haruspex

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    That's the speed of A just before the string becomes taut again. Yes, they will have the same speed as each other afterwards (what is the name for such an interaction?), but it won't be the same as the speed A had.
     
  6. Feb 12, 2019 at 10:20 PM #5
    I do not know. Is there a name for such interaction? Action - reaction?

    If speed of B when leaving ground ≠ speed of A when the string becomes taut again, how can we find speed of B? I don't think knowing the deceleration helps

    Thanks
     
  7. Feb 12, 2019 at 10:56 PM #6

    Delta2

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    You have to use conservation of momentum i think (though there are external forces which do not cancel out, still i think conservation of momentum can be applied cause the "collision" forces when string becomes suddenly taut are much more bigger than the external forces) to find the common speed of A and B when string becomes taut due to the movement of A.
     
  8. Feb 13, 2019 at 12:04 AM #7

    haruspex

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    If an interaction between moving bodies results in their moving together at the same velocity it is called a coalescence. What happens with these two masses when the string tautens is slightly different because they will move at the same speed but in opposite directions.
    As I wrote, the speeds will be the same after it is taut again, but not before.

    @Delta2 is correct, but as D2 notes it is not quite kosher just to say 'conservation of momentum'. I'll try to show you the authorised version:
    As far as A is concerned, what makes it suddenly change speed? (It doesn't "know" about B.) Likewise, what makes B start moving again?
     
  9. Feb 13, 2019 at 1:28 AM #8
    The external forces you mean are normal force and weight?

    A suddenly changes speed because it is being pulled upwards by the taut string? And same reasoning why B starts moving again, because it is pulled upwards by the taut string?

    I still do not know how to use conservation of momentum in this case. What is the system? The momentum between B and the string conserved?

    Thanks
     
  10. Feb 13, 2019 at 1:41 AM #9

    haruspex

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    Yes. There is a sudden large tension in the string for a short time, leading to an impulse (change in momentum) of each mass.
    What do you usually assume about tension in a string going over a pulley? What do you deduce about the two impulses?
     
  11. Feb 13, 2019 at 1:52 AM #10

    Delta2

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    it should be clear that the two masses exchange momentum each time the string becomes taut due to the movement of A downwards.
    The only technicality is that the two bodies have opposite velocities exactly after the string has become taut. You can bypass this and apply conservation of momentum like it was an inelastic collision and both bodies move with common velocity V just after the string has become taut.
     
  12. Feb 13, 2019 at 2:26 AM #11

    haruspex

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    Ah, but they do not exchange momentum. That would imply momentum is conserved, which it is not.
    I think it is instructive to go through what actually happens.
     
  13. Feb 13, 2019 at 2:34 AM #12

    Delta2

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    Strictly speaking you are correct , momentum is not conserved because the two bodies have same speed but opposite velocities, however as i said we can bypass this technicality and treat the situation as an inelastic collision where both bodies move with the same velocity V after the collision. The direction of V for each body is not important and it doesn't play a role to what this exercise asks.
     
  14. Feb 13, 2019 at 2:58 AM #13

    BvU

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    We are given that the string is inextensible. As far as I can see we (your helpers and you) don't have a common picture of what happens. But we agree on momentum conservation, so perhaps treating this events as a kind of collision (this time elastic) between an m with a certain speed and a 2m at rest would be a good way to proceed.

    I wonder if the exercise author meant you to go through all this, I find it getting complicated... :nb)
     
  15. Feb 13, 2019 at 3:39 AM #14

    haruspex

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    Elastic? That would be work conservation, but I assure you that is not conserved.
    Also, momentum is not conserved in the standard sense. There's a sign flip.
     
  16. Feb 13, 2019 at 4:39 AM #15

    ehild

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    The situation can be considered as curvilinear motion along the string, clockwise or ant-clockwise. When the string is taut, the velocities are along the tangent of the string and both bodies move clockwise or anti-clockwise, with the same speed. You can consider anti-clockwise motion positive, clockwise motion negative, and the momenta of both add in this sense, and are conserved during the "collision" when the bodies just start to interact through the string.
     
  17. Feb 13, 2019 at 6:06 AM #16

    jbriggs444

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    I doubt that the questioner intended for the string to come taut elastically. But neither is it ruled out by the problem statement. The adjective "inextensible" is not adequate to determine whether mechanical energy is conserved.
     
  18. Feb 13, 2019 at 1:59 PM #17

    haruspex

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    Interesting...
    In my experience, the standard intent in problems where an inextensible string becomes suddenly taut is that it is treated as a coalescence, i.e. the string is inelastic and the bodies attached immediately match speeds. But I had never noticed this is illogical and inconsistent with the way we treat collisions of incompressible bodies.
    Indeed, inextensible ought to imply perfectly elastic. If so, the string only becomes taut momentarily (in the British English sense).

    In the present problem that is going to complicate matters. It is not immediately apparent what will happen next - will the string become taut again before B hits the ground again? So I strongly suspect the intent is inelastic, but I will see what happens in the other view.
     
  19. Feb 13, 2019 at 2:09 PM #18

    jbriggs444

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    In my view, "inextensible" and "incompressible" are both orthogonal to "elastic". They provide no information either way. You end up with an indeterminate form, an infinite force integrated over a zero displacement either way. The whole point of elasticity or a coefficient of restitution is so that you can parameterize the loss of mechanical energy independent of the rigidity of the material.

    At the risk of over-generalizing, I suspect that instructors have an intuition for steel balls which are both very rigid and also very elastic and sometimes use this to leap from the idea that since an incompressible substance cannot deform and dissipate non-zero energy under a finite force that it therefore cannot dissipate energy at all, even if it encounters the proverbial irresistable force.

    Edit: By contrast, the intuition for inextensible cords tends to take hemp ropes as the model. Highly inextensible and highly inelastic.
     
    Last edited: Feb 13, 2019 at 2:58 PM
  20. Feb 13, 2019 at 2:22 PM #19

    haruspex

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    We certainly have to assume one extreme or the other or there is insufficient data to solve the problem.
    Because assuming perfect elasticity creates a complicated motion, I feel certain the intent here is inelastic.
    I found an article on JSTOR, https://www.jstor.org/stable/3616194?read-now=1&seq=3#metadata_info_tab_contents, which argues the opposite interpretation, but I think I see a flaw in that. If I am right, it leads to your view that extensibility is orthogonal to elasticity.

    The JSTOR article also corroborates my experience that such problems intend coalescence.
     
  21. Feb 13, 2019 at 4:11 PM #20

    BvU

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    At least | momentum |
    conservation. But I wouldn't know where the kinetic energy should go... The pulley does no work.
     
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