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(A) The time period of a vibration magnetometer is given by $T = 2 \pi \sqrt{\frac{I}{MH}}$,where $I$ is the moment of inertia and $M$ is the magnetic

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$5: 4$

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(A) The time period of a vibration magnetometer is given by $T = 2 \pi \sqrt{\frac{I}{MH}}$,where $I$ is the moment of inertia and $M$ is the magnetic moment.When like poles are joined,the effective magnetic moment is $M_1 + M_2$. Given $T_1 = 5 \, s$,we have $5 = 2 \pi \sqrt{\frac{I}{M_1 + M_2}}$.When unlike poles are joined,the effective magnetic moment is $M_1 - M_2$. Given $T_2 = 15 \, s$,we have $15 = 2 \pi \sqrt{\frac{I}{M_1 - M_2}}$.Dividing the two equations: $\frac{5}{15} = \sqrt{\frac{M_1 - M_2}{M_1 + M_2}} \Rightarrow \frac{1}{3} = \sqrt{\frac{M_1 - M_2}{M_1 + M_2}}$.Squaring both sides: $\frac{1}{9} = \frac{M_1 - M_2}{M_1 + M_2}$.Cross-multiplying: $M_1 + M_2 = 9M_1 - 9M_2 \Rightarrow 10M_2 = 8M_1$.Therefore,the ratio $\frac{M_1}{M_2} = \frac{10}{8} = \frac{5}{4}$.

Two different magnets are tied together and allowed to vibrate in a horizontal plane. When their like poles are joined,the time period of oscillation is $5 \, s$,and when their unlike poles are joined,the time period of oscillation is $15 \, s$. The ratio of their magnetic moments is

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