Two bar magnets ' $\mathrm{P}$ ' and ' $\mathrm{Q}$ ' are kept in uniform magnetic field 'B' with magnetic…

Two bar magnets ' $\mathrm{P}$ ' and ' $\mathrm{Q}$ ' are kept in uniform magnetic field 'B' with magnetic moments ' $\mathrm{M}_{\mathrm{P}}$ ' and ' $\mathrm{M}_{\mathrm{Q}}$ ' respectively. Magnet ' $\mathrm{P}$ ' is oscillating with frequency twice that of magnet ' $\mathrm{Q}$ '. If the moment of inertia of the magnet ' $\mathrm{P}$ ' is twice that of magnet ' $\mathrm{Q}$ ' then
  1. $\mathrm{M}_{\mathrm{Q}}=2 \mathrm{MP}_{\mathrm{P}}$
  2. $\mathrm{M}_{\mathrm{P}}=2 \mathrm{M}_{\mathrm{Q}}$
  3. $\mathrm{M}_{\mathrm{P}}=8 \mathrm{M}_{\mathrm{Q}}$
  4. $\mathrm{M}_{\mathrm{Q}}=8 \mathrm{M}_{\mathrm{P}}$

Solution

The frequency of oscillation given by $\begin{aligned} & \mathrm{f}_{\mathrm{p}}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{M}_{\mathrm{p}} \mathrm{B}}{\mathrm{I}_P}} \text { and } \mathrm{f}_{\mathrm{Q}}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{M}_{\mathrm{Q}} \mathrm{B}}{\mathrm{I}_{\mathrm{Q}}}} \\ & \mathrm{f}_{\mathrm{p}}=2 \mathrm{f}_{\mathrm{Q}} \\ & \frac{1}{2 \pi} \sqrt{\frac{\mathrm{M}_{\mathrm{P}} \mathrm{B}}{\mathrm{I}_{\mathrm{P}}}}=2 \cdot \frac{1}{2 \pi} \sqrt{\frac{\mathrm{M}_{\mathrm{Q}} \mathrm{B}}{\mathrm{I}_{\mathrm{Q}}}} \\ & \frac{\mathrm{M}_{\mathrm{P}}}{\mathrm{I}_{\mathrm{P}}}=4 \frac{\mathrm{M}_{\mathrm{Q}}}{\mathrm{I}_{\mathrm{Q}}} \\ & \therefore \frac{\mathrm{M}_{\mathrm{P}}}{\mathrm{M}_{\mathrm{Q}}}=4 \frac{\mathrm{I}_{\mathrm{P}}}{\mathrm{I}_{\mathrm{Q}}}=8 \quad\left[\because \mathrm{I}_{\mathrm{P}}=2 \mathrm{I}_{\mathrm{Q}}\right] \end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 2)

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