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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=[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]

## 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