A combination of two magnets perform $10$ oscillations per minute with similar poles together and $6$ oscillation per min with dissimilar poles together. The ratio of magnetic moments is.

8 : 17

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7 : 16

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5 : 18

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17 : 8

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Solution

The correct option is C $17 : 8$
The time period of a magnetic dipole is given by
$T = 2 π \sqrt{\frac{I}{M B_{H}}}$
Here, $I$ is the moment of Inertia, $M$ is the magnetic moment and $B_{H}$ is the horizontal component of Earth's magnetic field.

When two magnets are linked together, let $I^{'}$ be the new moment of inertia.

Let $M_{1}$ , $M_{2}$ be the magnetic moments of individual magnets.
If similar poles are together, new magnetic moment of the combined system is $M_{1} + M_{2}$
If dissimilar poles are together, new magnetic moment of the combined system is $M_{1} - M_{2}$

Therefore, $\frac{T 1}{T 2} = \sqrt{\frac{M_{1} + M_{2}}{M_{1} - M_{2}}}$
$\Rightarrow (\frac{10}{6})^{2} = \frac{M_{1} + M_{2}}{M_{1} - M_{2}} \Rightarrow \frac{M_{1}}{M_{2}} = \frac{25 + 9}{25 - 9} = \frac{17}{8}$

Thus, $M_{1} : M_{2} = 17 : 8$

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A combination of two magnets perform 10 oscillations per minute with similar poles together and 6 oscillation per min with dissimilar poles together. The ratio of magnetic moments is.

A combination of two magnets perform $10$ oscillations per minute with similar poles together and $6$ oscillation per min with dissimilar poles together. The ratio of magnetic moments is.