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Moment of inertia of a wire bent into a ring or a semicircle

This type of question, asked 3 times: 2021 Paper 10, 29 Jun 2022, 8 Apr 2023. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 10

    6
    Consider a uniform wire of mass MM and length L. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is:

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  2. Question 2

    29 Jun 2022

    117
    The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is I1I_{1}. The same rod is bent into a ring and its moment of inertia about a diameter is I2I_{2}. If I1I2\frac{I_{1}}{I_{2}} is xπ23\frac{x\pi^{2}}{3}, then the value of xx will be

  3. Question 3

    8 Apr 2023

    24
    The moment of inertia of semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is 1XMR2\frac{1}{X}MR^{2}, where RR is the radius and MM is the mass of semicircular ring. The value of xx will be