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Kinetic Energy rotational and translational conceptual question.

  1. Nov 15, 2012 #1
    1. The problem statement, all variables and given/known data

    A ball rolls without slipping down incline A, starting from
    rest. At the same time, a box starts from rest and slides down incline B, which
    is identical to incline A except that it is frictionless. Which arrives at the bottom
    first? (a) The ball arrives first. (b) The box arrives first. (c) Both arrive at
    the same time. (d) It is impossible to determine.

    2. Relevant equations

    KE = 1/2 I ω2

    KE = 1/2 m v2

    3. The attempt at a solution

    I answered D because i felt that you would need to know how much work had been done by friction. The correct answer is B however, and i'm not certain why. Does the ball rolling down the incline have both rotational and translational energy?
     
  2. jcsd
  3. Nov 15, 2012 #2
    Yes, the ball has got both translational and rotational energy. It is the extra term (the rotational energy) that makes the ball reach the "bottom" second. This is a result of the first law of thermodynamics, which states that energy cannot be created or destroyed. :)
     
  4. Nov 15, 2012 #3
    But wouldn't that mean it has more kinetic energy therefore it would be moving faster?
     
  5. Nov 15, 2012 #4
    No, the rotational energy and the translational energy must share the amount of energy equal to the decrease in potential energy.
     
  6. Nov 15, 2012 #5
    For the ball:

    potential energy --> kinetic energy + rotational energy (both are nonnegative quantities).

    Thus, the translational energy would be greater in the absence of the rotational energy.


    For the box, however: potential energy --> kinetic energy + rotational energy (where the rotational energy evidently is zero.) Thus, "maximal" kinetic energy.

    Since the mass cancels out in both cases, the latter case involves a translational energy that is greater per unit mass and thus the box travels at a higher speed.
     
  7. Nov 15, 2012 #6
    ok so basically the rotational + translation in the rolling ball would equal the translational of the sliding block, however because the rolling ball is losing energy to friction the box would be faster?
     
  8. Nov 15, 2012 #7

    haruspex

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    Well, no, it's not losing energy to friction exactly. The friction is causing some the energy to go into rotational KE instead of linear KE. So the linear acceleration is less than if the ball were also on a frictionless surface. If the slopes were to level out then rise again, the two objects would reach the same height, but the ball would take longer to get there.
     
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