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 is _ _ _ _ .

Step-by-step solution

For solid cylinder: I = 1 2 mR^2 Large cylinder: mass m₁ = R^2 L , so I₁ = 1 2 R^2 L R^2 = R^4 L 2 Small cylinder (carved out): length L/2, radius R/3, mass m₂ = (R/3)^2(L/2) = R^2 L 18 I₂ = 1 2 R^2 L 18 (R/3)^2 = 1 2 R^2 L 18 R^2 9 = R^4 L 324 I₁ I₂ = R^4 L/2 R^4 L/324 = 324 2 = 162