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Four bodies all have the same mass M and the same radius 2R: I₁ is the moment of inertia of a solid sphere about its diameter, I₂ that of a solid cylinder about its axis, I₃ that of a solid circular disc about its diameter, and I₄ that of a thin circular ring about its diameter. If 2(I₂+I₃)+I₄=x I₁, then the value of x will be ______.

Asked in JEE Main 25th June 2nd Shift 2022 · Scaling and comparison of moments of inertia

Answer: 5

Step-by-step solution

Every body has the same mass M and the same radius 2R, so write u=M(2R)² and express all four as multiples of it.

I₁=2/5u (solid sphere, diameter).

I₂=1/2u (solid cylinder, own axis).

I₃=1/4u (disc, diameter).

I₄=1/2u (ring, diameter — half of the ring's MR² about the perpendicular axis).

Left-hand side: 2(1/2+1/4)u+1/2u=3/2u+1/2u=2u.

So x·2/5u=2u, giving x=5.

The 2R appears in every term and cancels; only the shape factors matter.

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