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## Homework Statement

A uniform circular cylinder of mass `m' (a yo-yo) has a light inextensible string wrapped around it so that it does not slip. The free end of the string is fastened to a support and the yo-yo moves in a vertical straight line with the straight part of the string also vertical. At the same time the support is made to move vertically having upward displacement [itex]Z(t)[/itex] at time `t'. Find the upward acceleration of the yo-yo.

## Homework Equations

Define `a' as the radius of the cylinder.

Define theta as the rotation angle of the yo-yo.

[tex]V = -mg(a\theta - Z)[/tex]

[tex]T = 3/4 m v^2 = 3/4 m (a \dot{\theta} - \dot{Z})^2[/tex]

[tex]L = T-V[/tex]

Use, [tex]\frac{d}{dt} ( \frac{dL}{d\dot{\theta}} ) - \frac{dL}{d\theta} = 0[/tex]

## The Attempt at a Solution

[tex]\frac{d}{dt} ( \frac{dL}{d\dot{\theta}} ) = 3/2m(a\ddot{\theta} - \ddot{Z})[/tex]

[tex]\frac{dL}{d\theta} = mga[/tex]

So solving gives me,

[tex] \ddot{\theta} = \frac{2/3g +\ddot{Z}}{a} [/tex]

(downwards angular acceleration.)

Therefore upwards acceleration of the yo-yo is,

[tex] \ddot{z} = -a \ddot{\theta} = \ddot{Z}-2/3g[/tex].

But I'm missing a factor of 1/3 in front of [tex]\ddot{Z}[/tex] or,

[tex]\ddot{z} = 1/3(\ddot{Z}-2g)[/tex].

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