JEE Main 2021 · 25 July · morning · Single correct · +4 / −1

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Moment of inertia of a circular disc of mass 'M' and radius 'R' about X, Y axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be Iₓ, I y and I z respectively. The respectively radii of gyration about all the three axes will be the same.
Reason R : A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below :

  1. (a) Both A and R are correct but R is NOT the correct explanation of A.
  2. (b) A is not correct but R is correct.
  3. (c) A is correct but R is not correct.
  4. (d) Both A and R are correct and R is the correct explanation of A.
Answer and solution

Answer (b) A is not correct but R is correct.

Solution

Physics sir · 3 steps

Same disc, same mass, but different axes give different radii of gyration. Check I_z against Iₓ.

  1. Perpendicular axis theorem. For a flat disc, I_z = Iₓ + I_y. By symmetry Iₓ = I_y, and I_z = 1/2MR², so Iₓ = I_y = 1/4MR².

    k = √(I/M)

    AxisMoment of inertiaRadius of gyration
    X (in the plane)1/4MR²R/2
    Y (in the plane)1/4MR²R/2
    Z (perpendicular)1/2MR²R/√2
  2. Check each statement. kₓ = k_y, but k_z is larger, so the radii are not all the same and A is wrong. R is true: a rigid body keeps its mass and shape while it rotates.
  3. So. A is not correct but R is correct, which is option (b).

The idea to keep. Radius of gyration depends on the axis, not just on mass and shape. Find I about that axis first, then use k = √(I/M).

Answer from the NTA final answer key. Solution by PrepWiser.

Study Rotational Motion

Asked about 180 times in JEE Main since 2020. This question is from 2021.