Question

The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is $2sec$. The magnet is cut along its length into $3$ equal parts and the $3$ parts are then placed on each other with their like poles together. The time period of this combination will be

• A. $2sec$
• B. $\frac{2}{9}sec$
• C. $2\sqrt{3}sec$
• D. $\frac{2}{\sqrt{3}}sec$

Answer 1

Answer: (B)

$\frac{2}{9}sec$

$\frac{T_1}{T_2}=\sqrt{\frac{I_1}{I_2}}\sqrt{\frac{M_2}{M_1}}........\left(1\right)$
When the magnet is cut into three pieces, pole strength of the cut pieces is same as the original pole strength.
$I_1=\frac{m{l}^2}{12}$
$I_2=\frac{\frac{m}{3}(\frac{l}{3}{)}^2}{12}$
$\therefore =\frac{I_1}{I_2}=\frac{27}{1}$
$M_1=pole\times l$ where pole is the pole strength.
$M_2=pole\times \frac{l}{3}$
$\frac{M_2}{M_1}=3$
Eqn$\left(1\right)$ becomes
$\frac{T_1}{T_2}=\sqrt{\frac{27}{1}}\sqrt{3}=9$
Given $T_1=2$
$\therefore T_2=\frac{2}{9}$