A solid uniform cylinder of radius \(r\) rolls without sliding along the inside the surface of a hollow cylinder of radius \(R\), performing small oscillations. Determine time period.

**Solution:**

Translational kinetic energy + rotational kinetic energy + potential energy= constant

$$ \frac{1}{2}mv^2+{\frac{1}{2}I\omega^2+mg(R-r)(1-cos\theta)}=C$$

Now $$ I=1/2mr^2$$

$$ 3/4m(dx/dt)^2+mg(R-r)\theta^2/2=C$$

Differentiating with respect to time,

$$ \frac{3}{2}m(\frac{d{^2}x}{dt{^2}})^+mg(R-r)\theta\frac{d\theta}{dt} $$

Now, $$ x=(R-r)\theta$$

$$ \frac{3}{2} d^2x/dt^2(R-r)d\theta/dt+gxd\theta/dt=0$$

Cancelling \(\frac{d\theta}{dt}\) throughout

$$ \frac{d^2x}{dt^2}+\frac{2}{3}\frac{gx}{R-r}=0$$

this is the equation for SHM, with

$$ \omega^2=\frac{2}{3}\frac{g}{R-r}

$$

$$ T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{3(R-r)}{2g}}$$

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