Question

The length of a bar magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is $2s$. 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 :

• A. $2s$
• B. $2/3s$
• C. $2\sqrt{3}s$
• D. $2/\sqrt{3}s$

Answer 1

The correct option is B $2/3s$

The time period of oscillations of magnet
$T=2\pi \sqrt{\left(\frac{I}{MH}\right)}$ .....(i)
where $I=$ moment of inertia of magnet
$=\frac{m{L}^2}{12}$ ($m$, being the mass of magnet)
$M=$ pole strength $\times L$
and $H=$ horizontal component of earth's magnetic field.
When the three equal parts of magnet are place on one another with their like poles together, then
$I^{\prime }=\frac{1}{12}\left(\frac{m}{3}\right)\times {\left(\frac{L}{3}\right)}^2\times 3$
$=\frac{1}{12}\frac{m{L}^2}{9}=\frac{I}{9}$
and $M^{\prime }=polestrength\times \frac{L}{3}\times 3=M$
Hence, $T^{\prime }=2\pi \sqrt{\left(\frac{I/9}{MH}\right)}$
$\Rightarrow T^{\prime }=\frac{1}{3}\times T$
$T^{\prime }=\frac{2}{3}s$