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Classical Mechanics: Lagrangian of a falling yo-yo

  1. Feb 7, 2008 #1
    1. The problem statement, all variables and given/known data
    A uniform circular cylinder (a yo-yo) of radius a and mass M has a string wrapped around
    it that can unwind without slipping. The yo-yo moves in a vertical straight line and the
    straight part of the string is vertical as well. The other end of the string is fastened to a
    support that has upward displacment h(t) at time t. Here h(t) is a prescribed function, not
    a degree of freedom.
    a. Take the rotation angle φ of the yo-yo as a generalized coordinate and find Lagrange’s equation.
    b. Find the acceleration of the yo-yo. What must the support’s acceleration h(t) be so
    that the centre of the yo-yo can remain at rest?
    c. Suppose the system starts from rest. Find an expression for the total energy E = T+V at time t, in terms of h and h.

    2. Relevant equations
    L=T-V
    T=[tex]\frac{1}{2}[/tex]M[tex]\dot{y}[/tex][tex]^{2}[/tex]+[tex]\frac{1}{2}[/tex]I[tex]\dot{\varphi}[/tex][tex]^{2}[/tex]
    Defining downwards as my positive direction: V=-Mgy
    Rolling without slipping ---> a[tex]\dot{\varphi}[/tex]=[tex]\dot{y}[/tex]
    Since the support is moving upwards given by (h(t)) ---> y=[tex]\widetilde{y}[/tex]-h(t) where [tex]\widetilde{y}[/tex]= the distance between the yo-yo and the support at time t. So, [tex]\dot{y}[/tex]=[tex]\dot{\widetilde{y}}[/tex]-[tex]\dot{h(t)}[/tex]

    3. The attempt at a solution
    With these equations and treating [tex]\varphi[/tex] as my generalized coordinate, it's easy to obtain the Lagrange's equation of motion for this system. I get [tex]\dot{}\dot{\varphi}[/tex]=2/3*g/a

    Now for part b) of the question, I have [tex]\dot{}\dot{y}[/tex]=[tex]\dot{}\dot{h}[/tex]. If this is correct, I can figure this out using the without slipping equation above.
    From here on before starting part c), I am wondering why the Lagrange's equation I got is independent of h(t) (i.e its the same as if the support was fixed). And if its wrong, it has to do something with the relation between y and h(t). Any help is apprecaited
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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