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Motion of mass constrained to rail

  1. Jan 5, 2014 #1
    I have a classical mechanics problem related to the constrained motion of a mass:

    as per the attached graphics, assume a thin rigid rod of length L and mass m moving frictionless constrained to a rail such that is always locally perpendicular to the latter. The rod starts off with speed v=v0 in the straight path of the rail. My question is, what is the speed in the apex of the curved section (half a circle with radius R), and what is the speed after it is in the straight section again? There are no external forces acting.

    Thanks.

    attachment.php?attachmentid=65359&stc=1&d=1388939802.gif
     

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  2. jcsd
  3. Jan 5, 2014 #2

    A.T.

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    If it is frictionless and the rail is immobile then the velocity of the rod center is constant.
     
  4. Jan 5, 2014 #3
    There must be an external force on the rod. ( A central reaction force from the rail acting inward on the thin rigid rod. )
     
  5. Jan 5, 2014 #4

    mfb

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    In the curve, the motion along the rail is not the only contribution to the motion - rotation will need some energy.
    The velocity is lower, but goes back to the initial value once the rod leaves the curve - this is a result of energy conservation.

    The force between rod and rail is internal.
     
  6. Jan 5, 2014 #5

    Dale

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    You have ##KE=\frac{1}{2} m v^2 + \frac{1}{2} I \omega^2## during the curve. From that you can do a little algebra to get:
    [tex]v=\frac{2\sqrt{3}R}{\sqrt{L^2+12 R^2}}v_0[/tex]
     
    Last edited: Jan 5, 2014
  7. Jan 5, 2014 #6
    But the rod could only be slowed down and accelerated by a force along the rail (after all, the motion is essentially restricted to one degree of freedom). Where does this force come from?

    Also, wouldn't the rod have to be slowed down and sped up instantaneously to the appropriate speed when it enters and leaves the curved section?
     
    Last edited: Jan 5, 2014
  8. Jan 5, 2014 #7

    Dale

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    It comes from whatever it is that accomplishes this:
     
  9. Jan 5, 2014 #8

    mfb

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    Yes, in the same way your rail needs "infinite forces" to start the rotation. A more realistic setup would have a curvature that increases gradually, then the forces and accelerations stay finite.
     
  10. Jan 5, 2014 #9
    Wrong, I think.

    ##KE=\frac{1}{2} m v^2 ##

    or

    ##KE=\frac{1}{2} I \omega^2##

    whatever you choose. You can not apply both formulae to a spinning mass.

    It's common sense that the velocity of the rod center is constant.
     
  11. Jan 5, 2014 #10

    mfb

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    Wrong, you have to account for both contributions (or choose a frame where the mass is not moving, only rotating - the center of the curve).
     
  12. Jan 5, 2014 #11

    AlephZero

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    If you want the problem to be physically realistic (no "infinite forces" or "finite impulses applied instantaneously") then the track much have a continuously varying radius of curvature. Also, to apply a moment to the rod, there must be at least two points of contact, at different positions along the length of the track.

    With those assumptions, the curvature at the different points of contact will be different, and the resultant forces on the rod can have a component that changes the "linear" speed of the rod along the track, as well as changing its angular velocity.
     
  13. Jan 5, 2014 #12

    sophiecentaur

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    I don't think that would be necessary because the rod itself, by taking time to enter the curve, would not need an infinite impulse to get it turning. The front section would be deflected a bit, on entering the curve and the deflection would reach a maximum once the rear section was on the curve. (It's a low pass temporal filter, in effect). But, in any case, it's only a step change in acceleration, which doesn't imply an infinite force - just a step change in force.
    I don't think the angular rotation is any more relevant than the lateral deflection of the path. With an infinitely rigid rail and infinite mass, no energy can be lost from the rod.
     
  14. Jan 5, 2014 #13

    AlephZero

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    We can have different interpretations of exactly how to turn this into a physically realistic problem, but the bottom line is, the rod must be able to move along the track without any step changes in velocity (either linear or angular).

    I don't see why do you need "infinite mass" anywhere, unless you really meant "the track is fixed in an inertial coordinate system".
     
  15. Jan 5, 2014 #14

    Dale

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    Last edited by a moderator: May 6, 2017
  16. Jan 5, 2014 #15
    I don't understand why there is so much shock over the concept that, in the problem as stated, the forces and moments at the instantaneous transition between the straight and curved sections have to be infinite. We deal with this all the time in problems involving impact. In this problem, we basically have impact forces and moments applied by the rail at the two transitions. Big deal.

    Chet
     
  17. Jan 5, 2014 #16

    AlephZero

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    I agree with that in principle, but the "big deal" is that the OP's question about the velocity of the rail is not answerable as an impact problem, unless you assume something about the energy loss during the impact.

    And you still need to explain away how the the impact can change the rod's angular velocity, without friction forces acting.

    But I suppose whoever invented the problem expected the "average student" to realize you could get an answer using conservation of energy and just do the algebra, instead of thinking too hard about whether that method was justifiable.
     
  18. Jan 6, 2014 #17

    sophiecentaur

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    I do. I was just putting it in a practical context.
     
  19. Jan 6, 2014 #18
    I think I can see how the impact can change the rod's angular velocity without friction acting. There is a "moment impulse" also. As you yourself indicated, there will be 2 points of contact in the curved section. Going into the curved section, there will first be one point of contact, and the, in rapid succession, a second point of contact. These normal contact forces will be very high at the transition, and produce the impulse moment. What is your opinion? Does this make any kind of sense?

    Chet
     
  20. Jan 6, 2014 #19

    sophiecentaur

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    Surely, as the rod goes round the curve, its tangential velocity must be less than when along the straight. This allows the total KE (Rotational + Translational) to be unchanged. There will be a compression force on the rod as it enters the curve (slowing it down (gently - as the whole transitions onto the curve; there needs to be no step function in force, I think) and a tension, speeding it up on exit.
    We needn't have friction if we don't want it and neither do we need any instant changes of anything.
     
  21. Jan 6, 2014 #20

    Dale

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    I agree completely. The problem as stated included a bunch of simplifying but unrealistic assumptions (thin, rigid, completely constrained). Those assumptions can be understood as approximations. Under those approximations you get an impulsive torque at either end. That impulsive torque is also unrealistic, but can similarly be understood as an approximation. If you want to get rid of it then you merely have to get rid of the unrealistic approximations that generated it.
     
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