Problem — Ranking the moments of inertia of three cylinders

Problem

Three uniform cylinders of the same mass and the same radius: A solid cylinder; B thick-walled hollow cylinder (inner radius $R/2$); C thin ring (all the mass at radius $R$). Rank the moments of inertia about the cylinder’s axis, from smallest to largest.

Solution

The moment of inertia grows the further the mass is from the axis. For equal mass and outer radius:

• A (solid): mass distributed from $0$ to $R$, $I_A = \tfrac{1}{2}MR^2$;
• B (hollow, $R_\text{int}=R/2$): mass distributed from $R/2$ to $R$, $I_B = \tfrac{1}{2}M(R_\text{int}^2 + R^2) = \tfrac{1}{2}M(\tfrac{R^2}{4}+R^2) = \tfrac{5}{8}MR^2$;
• C (ring): all the mass at $R$, $I_C = MR^2$.

So $\tfrac{1}{2} < \tfrac{5}{8} < 1$: $\ev{I_A < I_B < I_C}$

Links

Topics: Rotational dynamics Concepts: Moment of inertia Skills: Ranking reasoning Objects: Rotating disc