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Frictionless Pulley with two weights

  1. Aug 21, 2009 #1
    Firstly, nice forum!
    Secondly, I am Swedish but I will try to use the correct scientific terms in explaining the problem and my proposed solution.

    1. The problem statement, all variables and given/known data
    There is a string of negligible mass which "hangs" over a pulley attached to a frictionless horizontal axis. On each end of the string there is a attached a weight. One of the weights (i've chosen the one on the right hand side if drawn the "normal way") is twice the mass of the weight of the other.
    The system starts out at rest.

    [pulley] = Radius R, and Moment of Inertia I = (1/5)*m*R².
    [weights] = A = m and B = 2m.
    [string tension] = left side (A): T ; right side (B): S.
    [gravity] = As always g denoted the pull of the earth (9.82 m/s² here in Sweden).

    When the weights are released from rest (and left to fall freely), what will be the tension on the string on the two sides of the pulley?

    2. Relevant equations
    In my attempted solution below I used:
    Newtons II equation where the sum of the forces are equal to the mass times the acceleration (F = m*a).
    I also used Eulers II law for momentum (torque) which in this case would be that the momentum is equal to the moment of inertia times the angular acceleration (M = I*alpha)
    I also used that alpha = R * a (along the x-axis (which I set to be positive upward))

    3. The attempt at a solution
    So, as listed above I will call the momentum about the z-axis of the pulley M, the angular acceleration alpha. I called the "left" pulley A and the "right" pulley B.
    I set the x-axis to the positive in the upward direction. And I assumed that the string was not elastic in any way.

    First I used newtons second law to describe the forces acting on pulley A to be
    F = T - mg = m * a
    and respectively for B:
    F = S - 2mg = m * a

    Then I used Eulers II with the moments origo in the center of the pulley to be
    M = RT - RS = I * alpha (and since alpha = R * a) = T-S = I * a.

    I then simply solved the equations using both NII and EII to get my expressions for what the tensions on S and T are.

    I got that T =m(a((R²/5)-1)+2g)
    and S = m(a(1-(R²/5))+g)

    It should be added I am at the moment of typing a little tired. Also I should probably have made a picture, but I hope my explanation of the problem will be enough.

    Usually I would be satisfied with my answer but the "hint" to the question was that I could use that acceleration (dv/dt) could be written as v*dv/ds (using the chain rule)! Why and how could that be used? I am now worried that I have made some very basic assumption or something wrong. Anyways help much appreciated!
     
  2. jcsd
  3. Aug 21, 2009 #2

    Doc Al

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    Staff: Mentor

    OK.
    Two errors on the right hand side:
    (1) You have m, but should have 2m.
    (2) Careful with signs: If the acceleration of A is +a, what must be the acceleration of B?

    Two problems:
    (1) Careful with signs. S > T, so use RS - RT = I*alpha.
    (2) alpha = a/R, not a*R.


    That hint doesn't seem to apply to this problem. No calculus is needed here.
     
  4. Aug 22, 2009 #3
    then the acceleration of B must of course be negative! thanks!

    So the new NII for B is: S - 2mg = -2ma

    Gah! stupid mistake! I misinterpreted some old notes of mine. Of course alpha = a/R!

    I am heading out to a wedding now (my girlfriends sister is getting married) I will be sure to post the completed solution when I get back tomorrow! Thank you so much for the help!
     
  5. Aug 22, 2009 #4

    ideasrule

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    Homework Helper

    Best of luck for your gf's sister and her husband!
     
  6. Aug 23, 2009 #5
    [pulley] = Radius R, and moment of inertia I = (1/5)mR².
    [weights] = A = m and B = 2m.
    [string tension] = left side (A): T ; right side (B): S.
    [gravity] = As always g denotes the pull of the earth (9.82 m/s² here in Sweden).

    The (all new!) attempt at a solution
    So, as listed above I will call the momentum about the z-axis of the pulley M, the angular acceleration alpha. I called the "left" pulley A and the "right" pulley B.
    I set the x-axis to the positive in the upward direction. And I assumed that the string was not elastic in any way. Also I have come to realise that the main reason to use Euler II in this question is to solve so we get an expression that doesn't assume that the acceleration is known, so that the tension of the strings will only depend on the weight (m) (and g) of the pulley and weights on the string.

    First I used Newtons second law to describe the forces acting on pulley A to be:
    F = T - mg = ma
    and respectively for B:
    F = S - 2mg = -2ma

    Then I used Eulers II law with the moments origo in the center of the pulley (my y-axis is positive going upwards, and the pulley is rotating clockwise!) to be:
    M = RT - RS = I * alpha (and since alpha = a/R) = R(T - S) = mR²a/(5R) = T-S=ma/5.

    I now solved the NII equations for T and S which made them:
    T = ma + mg and S = 2mg - 2ma
    In combination with EII:
    ma+mg - 2mg+2ma = ma/5
    Solving for a:
    15a-5g = a -> 14a = 5g -> a = 5g/14

    Thus I inserted the acceleration a into my equations for T and S:
    T = 5mg/14 + 14mg/14 = 19mg/14
    S = 28mg/14 - 10mg/14 = 9mg/7

    Could this be the correct answer?
     
  7. Aug 23, 2009 #6

    Doc Al

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    Staff: Mentor

    OK.
    You still have a sign problem; see my previous post.
     
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