Question

A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.

Answer 1

Let the angular velocity of the system about the point is suspension at any times be ${}^{\prime }\omega {.}^{\prime }$

$Spve=(R-r)\omega$

Again $ve=r\omega$

[where ${\omega }_1$= rotaional velocity of the sphere ]

$l=\frac{v_e}{r}=\left(\frac{R-r}{r}\right)\omega ...\left(1\right)$

By energy method, total energy in SHM is constant.

So, $mg(R-r)(1-cos\theta )+\frac{1}{2}m{v}_c^2+\frac{1}{2}I{\omega }^2=$ constant

$\therefore mg(R-r)(1-cos\theta )+\frac{1}{2}m(R-r{)}^2{\omega }^2+\frac{1}{2}m{r}^2\left(\frac{R-r}{r}\right){\omega }^2=$ constant

$\therefore mg(R-r)(1-cos\theta )+\frac{1}{2}m(R-r{)}^2{\omega }^2+\frac{1}{5}m{r}^2\left(\frac{R-r}{r}\right){\omega }^2=$ constant

$\Rightarrow g(R-r)(1-cos\theta )+(R-r){\omega }^2[\frac{1}{2}+\frac{1}{5}]=$ constant

Taking derivative,

$g(R-r)sin\theta \frac{d\theta }{dt}=\frac{7}{10}(R-r{)}^{2}2\omega \frac{d\omega }{dt}$

$\Rightarrow gsin\theta =2\times \left(\frac{7}{10}\right)(R-r)\alpha$

$\Rightarrow gsin\theta =\left(\frac{7}{5}\right)(R-r)\alpha$

$\Rightarrow \alpha =\frac{5gsin\theta }{7(R-r)}$

$=\frac{5g\theta }{7(R-r)}$

$\therefore \frac{\alpha }{\theta }={\omega }^2$

$=\frac{5g}{7(R-r)}=$ constant

So the motion is S.H.M. again

$\omega =\sqrt{\frac{5g}{7(R-r)}}$

$\Rightarrow T=2\pi \sqrt{\frac{7(R-r)}{5g}}$