Explanation

(b) In the first case, $T_{1} = \frac{60}{10} = 6$ s
$T_{1} = 2\pi\sqrt{\frac{I}{MH}} = 6$ …(i)
In the second case, $T_{2} = \frac{60}{14} = \frac{30}{7}$ s
If $B$ magnetic induction due to external magnet, then
$T_{2} = 2\pi\sqrt{\frac{I}{M(H + B)}} = \frac{30}{7}$ ….(ii)
Dividing Eq. (i) by Eq. (ii), we get
$\frac{6}{30/7} = \sqrt{\frac{H + B}{H}} = \sqrt{1 + \frac{B}{H}}$
or $\left( \frac{7}{5} \right)^{2} = 1 + \frac{B}{H}$
or $\ \frac{B}{H} = \frac{49}{25} - 1 = \frac{24}{25}$
If $n$ is number of vibrations/min in the third case when polarity of external magnet is reversed, then
$T_{3} = \frac{60}{n} = 2\pi\sqrt{\frac{I}{M(H - B)}}$ …(iii)
Dividing Eq. (i) by Eq. (iii), we get
$\frac{60}{60/n} = \sqrt{\frac{H - B}{H} = \sqrt{1 - \frac{B}{H}} = \sqrt{1 - \frac{24}{25}}}$
$\frac{n}{10} = \frac{1}{5},n = 2\ vibs - \min^{- 1}{.\ }$