A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to

Question

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. 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 $I_2$. The ratio $I_1 / I_2$ is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

The moment of inertia of a uniform solid cylinder about its central longitudinal axis is $I = \frac{1}{2}MR^2$. Expressing the total mass in terms of uniform density $ho$, $M = ho V = ho (\pi R^2 L)$. Consequently, the moment of inertia formulates as $I = \frac{1}{2}(ho \pi R^2 L) R^2 = \frac{1}{2}ho \pi L R^4$. For the original cylinder, $I_1 = \frac{1}{2}ho \pi L R^4$. For the carved-out inner cylinder, substituting $L ightarrow L/2$ and $R ightarrow R/3$ yields $I_2 = \frac{1}{2}ho \pi (L/2) (R/3)^4 = \frac{1}{2}ho \pi L R^4 	imes \frac{1}{2} 	imes \frac{1}{81} = I_1 	imes \frac{1}{162}$. Thus, the ratio $I_1 / I_2$ exactly evaluates to 162.