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Homework Help: Acceleration of a pulley

  1. Sep 20, 2010 #1
    1. The problem statement, all variables and given/known data

    A horizontal force F is applied to a frictionless pulley of mass m2. The horizontal surface is smooth. Show that the acceleration of the block of mass m1 is twice the acceleration of the pulley.

    LOOKS LIKE THIS: http://cnx.org/content/m14060/latest/npq1.gif
    But WITHOUT block B and its string.

    2. Relevant equations
    F=ma


    3. The attempt at a solution
    I drew separate force diagrams for m1 (the block) and m2 (the pulley. In the x direction, the block is only being acted on by T1 going in the pos. x direction. In the x direction, the pulley is being acted on by 2T1 and F. 2T1 is going in the neg. x direction. F, the opposite.

    I have to show that a1= 2a2

    So:

    The pulley is in equilibrium:
    F-2T1 = 0
    m2a2 - 2(m1a1)=0

    . . .and I don't know where to go from here. . . .I can't eliminate mass. . .


    Thank you in advance!
     
  2. jcsd
  3. Sep 20, 2010 #2
    It is easy to see that the pulley is moving. Let s_1 be the distance between pulley and A, s_2 be the distance between pulley and B. During the pull, s_2 stays constant. For s_1, it is decreasing right? But that section goes to the upper side of the pulley, so we can set 2s_1 equals also a constant. the reason of 2s_1 comes from the initial situation, we ignore the upper portion that is left of the originial position of A.

    Hence 2s_1+s_2=constant, differentiate twice yields your desired result.
     
  4. Sep 20, 2010 #3
    Oh. IDK why I thought the pulley was in equilibrium.

    Thank you so much!!
     
  5. Sep 20, 2010 #4
    I made the same mistake as you when I was having a introductory mechanics class.
    The method I present here is sometimes refered to no-stretch assumption. I don't know why the method is always not mentioned in the textbooks. Is it too obvious for the authors?
     
  6. Sep 20, 2010 #5
    I suppose they think so! The textbook (College Physics, 3rd ed. Serway & Faughn) says nothing about using distances and time derivatives . . . xD

    Thank you!
     
    Last edited: Sep 20, 2010
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