Question

A thin magnetic iron rod of length $30cm$ is suspended in a uniform magnetic field. Its time period is $4s$. If it is broken into three equal parts, the time period of oscillation of one part in seconds, when suspended in the same magnetic field is :

• A. $\frac{4}{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\sqrt{3}$
• D. $\frac{4}{\sqrt{3}}$

Answer 1

Answer: (A)

$\frac{4}{3}$

Time period of vibration: $T\propto \sqrt{\frac{I}{M}}$ where $I$ is the moment of inertia and M is the magnetic moment
$\therefore$ $\frac{T_1}{T_2}=\sqrt{\frac{I_1{M}_2}{I_2{M}_1}}$
When the magnet is broken into 3 pieces, magnetic moment becomes ${\frac{1}{3}}^{rd}$.
$\therefore M_1=3M_2$
Moment of inertial also changes.
$I_1=\frac{m{l}^2}{12}$
$I_2=\frac{\frac{m}{3}{\left(\frac{l}{3}\right)}^2}{12}$
$\therefore \frac{T_1}{T_2}=\sqrt{\frac{I_1{M}_2}{I_2\times 3M_2}}=\sqrt{\frac{1\times 27}{3}}=3$
$\therefore T_2=T_1\times \frac{1}{3}=\frac{4}{3}$