A compass needle oscillates 20 times per minute at a place where...

The formula for time period is given by

$T = 2\pi\sqrt{\dfrac{I}{M{B}_{H}}}$

$⇒ T = 2\pi\sqrt{\dfrac{I}{MBcos\theta}}$

Let

${\theta}_{1} = 30 ° {\theta}_{2} = 60 °$

The ratio of the time periods is,

$\dfrac{{T}_{1}}{{T}_{2}} = \dfrac{\dfrac{60}{20}}{\dfrac{60}{30}} = \sqrt{\dfrac{{B}_{2}\cos 60 °}{{B}_{1}\cos 30 °}} = \sqrt{\dfrac{{B}_{2}}{{B}_{1}} \times \dfrac{\dfrac{1}{2}}{\dfrac{\sqrt{3}}{2}}}$

$⇒ \dfrac{{B}_{1}}{{B}_{2}} = \dfrac{4}{\sqrt{243}}$