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Oscillation magnetometer: time period or oscillation count and the ratio of moments or fields

This type of question, asked 2 times: 27 Jul 2022, 12 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    27 Jul 2022

    12
    Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of 3 s3\text{ }s and 4 s4\text{ }s respectively. If their moments of inertia are in the ratio of 3:23:2 then the ratio of their magnetic moments will be:

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  2. Question 2

    27 Jul 2022

    102
    A compass needle of oscillation magnetometer oscillates 20 times per minute at a place PP of dip 30∘30^{\circ}. The number of oscillations per minute become 10 at another place QQ of 60∘dip60^{\circ}dip. The ratio of the total magnetic field at the two places (BQ:BP)\left( B_{Q}:B_{P} \right) is:

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  3. Question 3

    12 Apr 2023

    23
    A compass needle oscillates 20 times per minute at a place where the dip is 30∘30^{\circ} and 30 times per minute where the dip is 60∘60^{\circ}. The ratio of total magnetic field due to the earth at two place respectively is 4x\frac{4}{\sqrt{x}}. The value of xx is