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Work Done without using the Conservation of Energy

  1. Oct 11, 2008 #1
    1. The problem statement, all variables and given/known data

    A small mass m is pulled over the top of a frictionless half cylinder of radius R by a massless cord passing over the top. Without using the conservation of energy, find the work done in moving the mass from the bottom to the top of the cylinder at a constant speed, in terms of m, g, and R.

    2. Relevant equations

    ?


    3. The attempt at a solution

    Really not sure how to approach this... Can I somehow set up an integral?
     
  2. jcsd
  3. Oct 11, 2008 #2

    Doc Al

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    Staff: Mentor

    Yes, you'll need to set up an integral of F*ds along the cylinder. Figure out the force that must be applied as a function of position along the cylinder. Hint: Measure position in terms of angle, from 0 to ∏/2 radians.
     
  4. Oct 13, 2008 #3
    I understand the concept, but I am still having trouble setting up the function. I guess I can disregard normal force, so the only force doing work is the tension.

    So the work of tension is Force of T * displacement * cos angle. Since the tension is always in the same direction as the displacement, the angle will always be cos 0, or 1, so I can leave that out. The displacement will be R * angle. I don't know how to figure out the force of the tension at any given point since the weight is sometimes pulling down (like at 0 degrees) and sometimes at a right angle, where I think it should then be doing no work.
     
  5. Oct 14, 2008 #4

    Doc Al

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    Yes. We only care about the work done by the tension force.
    At any given position, the infinitesimal displacement will be Rdθ.
    Since the speed is constant, the tension must be just enough to balance out the tangential component of the weight. Find that as a function of θ.
     
  6. Oct 14, 2008 #5
    Not sure I know how to do that...

    I think the integral should be from 0 to pi/ 2 of Rdtheta * wcos theta?
     
  7. Oct 14, 2008 #6

    Doc Al

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    Exactly!

    Now just do that integral.
     
  8. Oct 14, 2008 #7
    Also: check your answer. you should be able to use conservation of energy to check it at least!

    I did a similar problem (frictionless half-pipe) with some AP teachers in outreach last month.
     
  9. Oct 14, 2008 #8
    So the work done is just wR [sin theta]0 to pi/2 =

    wR sin(pi/2) - wR sin(0) = wR ?
     
  10. Oct 14, 2008 #9
    Also, how can I check using the conservation of energy if I don't know the speed?
     
  11. Oct 14, 2008 #10

    Doc Al

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    Yep.
     
  12. Oct 14, 2008 #11

    Doc Al

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    The speed doesn't change.
     
  13. Oct 14, 2008 #12
    OK. I get it. Thanks to all of you for your help!
     
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