A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to I₁. A small co-centric cylinder of length L/2 and radius R/3 carved from this cylinder has moment of inertia about its axis equals to I₂. The ratio I₁/I₂ i s _____.
Physics · Rotational Motion · moment of inertia and kinetic energy in pure rotation
JEE Mains 2026 24 January Evening Shift

Question

A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to I₁. A small co-centric cylinder of length L/2 and radius R/3 carved from this cylinder has moment of inertia about its axis equals to I₂. The ratio I₁/I₂ i s _____.

Physics · Rotational Motion · moment of inertia and kinetic energy in pure rotation

JEE Mains 2026 24 January Evening Shift

G1 Physics · Accepted Answer

Let the original cylinder have mass M. Since mass is proportional to volume for the same material, m/M = [(R/3)²(L/2)]/ [R²L] = 1/18. Therefore the carved small cylinder has mass m = M/18. For a solid cylinder about its own axis, I = ½MR². Hence, I₁ = ½MR². For the small cylinder, I₂ = ½m(R/3)² = ½(M/18)(R²/9) = MR²/324. Therefore, I₁/I₂ = (MR²/2)/ (MR²/324) = 324/2 = 162. Hence the answer is 162.