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Which will reach the bottom first?

  1. Feb 12, 2015 #1
    hello

    let's say we have two cylindrical rodes of the same material and of the same length, but with different diameters

    we place them in an inclined floor and let them roll without friction

    which will reach the bottom first? the thick or the thin?

    my understanding for this question is that we need to calculate the torque that makes the rod roll, which is depended on the weight of the rod and the diameter of the rod

    then we need to calculate the angular velocity that this torque generates, which is depended on the circumference of each rod, along with the torque generated due to the mass/weight of each rod

    in my opinion, we cannot be sure that one or the other will move faster, if we don't do these calculations

    or I miss something?

    thanks!
     
  2. jcsd
  3. Feb 12, 2015 #2

    Svein

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    If there is no friction, why would they roll?
     
  4. Feb 12, 2015 #3
    ok, then only air resistance is negligible
     
  5. Feb 12, 2015 #4
    so without friction, they would be static? what about the torque generated by their weight? it would cause them to rotate endlessly without moving out of their position at all?
     
    Last edited: Feb 12, 2015
  6. Feb 12, 2015 #5

    jbriggs444

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    If you draw a free body diagram containing only the normal force and the force of gravity but no friction, where do you see a torque?

    Edit: In particular, a torque about the center of the cylinder.
     
  7. Feb 12, 2015 #6
    VFjeJ05.png

    so what is your opinion? that the rod will stay there immobilized?
     
  8. Feb 12, 2015 #7

    jbriggs444

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    Why do you draw the force of gravity as if it acts on the left edge of the cylinder?
     
  9. Feb 12, 2015 #8
    you can have it like that, but don't make the mistake to replace the whole rod with a point body
    KHQVX9J.png
     
  10. Feb 12, 2015 #9
    there is still a couple of forces exerted, which will make the rod to roll, given there is rolling friction, but not air resistance
    pKN41fh.png
     
  11. Feb 12, 2015 #10

    jbriggs444

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    I see a "free body diagram" with a force of gravity correctly positioned to act at the center of gravity of the cylinder but with two additional force vectors at right angles. What are those? And where is the normal force from the slope on the cylinder?

    I see a free body diagram with a force acting parallel to the downward slope acting on the center of the cylinder and a force in the opposite direction acting at the point of contact. What force is represented by the down-slope arrow? What force is represented by the up-slope arrow?
     
  12. Feb 12, 2015 #11

    A.T.

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    Do you mean there is static contact friction that makes it roll, but no rolling resistance that dissipates energy? If that's the case, then you can use energy conservation to find the speeds.
     
  13. Feb 12, 2015 #12
    exactly
    so, from energy conservation, we deduct that the larger mass will have the (much) lower velocity
    but, the potential energy is not quite the same at the begining, because the center of mass is a big higher in the larger mass
    so are there couples of cylinders (with different diameter) that the larger will be slower and other couples where the larger will be faster?
     
  14. Feb 12, 2015 #13
    you see the analysis of the force of gravity
    and in the second, you see the moving part of the weight force analysis, and its couple force at the point of friction
     
  15. Feb 12, 2015 #14

    A.T.

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    The change in height is the same for both cylinders, and the change in potential energy is all that matters.
     
  16. Feb 12, 2015 #15

    jbriggs444

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    Note that the force of gravity produces zero torque about the center of cylinder. The only force that produces a non-zero torque is the force of friction at the point of contact.

    Note that there is no guarantee that the force of friction is equal in magnitude to the resultant of gravity plus the normal force. To my mind, that means that referring to the two as a "couple" is not accurate.
     
  17. Feb 12, 2015 #16
    so what is the answer?
     
  18. Feb 12, 2015 #17

    jbriggs444

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    The point is not to give you the answer. The point is to get you to reason out the answer for yourself.
     
  19. Feb 12, 2015 #18
    I found the answer: the bigger mass the lowest the velocity, is this the case?
     
  20. Feb 12, 2015 #19

    jbriggs444

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  21. Feb 12, 2015 #20
    are they reaching the bottom the same time?
     
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