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NahsiN
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Homework Statement
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 displacement 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.
Homework Equations
L=T-V
T=\frac{1}{2}M\dot{y}^{2}+\frac{1}{2}I\dot{\varphi}^{2}
Defining downwards as my positive direction: V=-Mgy
Rolling without slipping ---> a\dot{\varphi}=\dot{y}
Since the support is moving upwards given by (h(t)) ---> y=\widetilde{y}-h(t) where \widetilde{y}= the distance between the yo-yo and the support at time t. So, \dot{y}=\dot{\widetilde{y}}-\dot{h(t)}
The Attempt at a Solution
With these equations and treating \varphi as my generalized coordinate, it's easy to obtain the Lagrange's equation of motion for this system. I get \dot{}\dot{\varphi}=2/3*g/a
Now for part b) of the question, I have \dot{}\dot{y}=\dot{}\dot{h}. 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