The length of a magnet is large compared to its width and breadth. The time period of its width and breadth.…
The length of a magnet is large compared to its width and breadth. The time period of its width and breadth. The time period of its oscillation in a vibration magnetometer is $2 \mathrm{~s}$. The magnet is cut along its length into three equal parts and three parts are then placed on each other with their like poles together. The time period of this combination will be
$2 \mathrm{~s}$
$2 / 3 \mathrm{~s}$
$2 \sqrt{3} \mathrm{~s}$
$2 / \sqrt{ } 3 \mathrm{~s}$
Solution
Time period of vibration, $T=2 \pi \sqrt{\frac{1}{M B}}$
Where $\ell=$ moment of inertia of magnet, $\mathrm{M}=$ magnetic moment $\mathrm{I}=\frac{\mathrm{m} \ell^2}{12}$ and $\mathrm{M}=$ pole strength $\times \ell$
$
I^{\prime}=\frac{1}{12}\left(\frac{m}{3}\right)\left(\frac{\ell}{3}\right)^2 \times 3=\frac{\mathrm{l}}{9}
$
and $M^{\prime}=$ pole strength (will remain the same) $\times(\ell / 3) \times 3=M$.
$
\mathrm{T}^{\prime}=\frac{\mathrm{T}}{\sqrt{9}}=\frac{2}{9} \mathrm{~s} .
$