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The yo-yo

  1. Oct 11, 2009 #1
    1. The problem statement, all variables and given/known data
    A yo-yo is made from two uniform disks, each with mass m and radius R, connected by a light axle of radius b. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the linear acceleration of the yo-yo.


    2. Relevant equations
    [itex]
    \tau=TR=I\alpha[/itex]

    [itex]
    F=ma[/itex]


    3. The attempt at a solution
    [itex]\tau=TR=I\alpha[/itex]

    [itex]Tb=2\left(\frac{1}{2}mR^2\right)\alpha[/itex]

    since [itex]a_{tan}=r\alpha[/itex],substituted into the equation above and simplified,

    [itex]
    Tb=mRa[/itex] ...1

    The yo-yo is accelerating downwards linearly, so

    [itex]
    2mg-T=2ma[/itex] ...2

    Solving for T in eq.1 and substituting into eq.2,

    [itex]
    2mg-\frac{mRa}{b}=2ma[/itex]

    Solving for a, I got

    [itex]
    a=\frac{2g}{2+R/b}[/itex]

    which is not the right answer... the correct answer is
    [itex]
    a=\frac{2g}{2+(R/b)^2}[/itex]

    what did I do wrong??
     
  2. jcsd
  3. Oct 11, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi kudoushinichi88! :smile:

    (I haven't actually checked your equations, but …)

    wouldn't it be easier to use conservation of energy? :wink:
     
  4. Oct 11, 2009 #3

    rl.bhat

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    Homework Helper

    String unwinds around the cylinder of radius b. So a(tan) = b
     
  5. Oct 11, 2009 #4
    You substituted the wrong value for r. If the yoyo unwinds with angular speed [tex]\omega[/tex]
    the vertical speed of the yoyo is [tex]b \omega[/tex] and not [tex]R \omega[/tex]
     
  6. Oct 13, 2009 #5
    ah... so that's why!
    Thank you all! This has also helped me to find the angular acceleration and tension in the string...
     
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