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 it has moment of inertia about its axis equal to \(I_2\). The ratio \(I_1/I_2\) is ________.

Hint:

For bodies of same material, mass ratios equal volume ratios.

Answer 1

Correct Answer: 162

Step 1: Moment of inertia of a solid cylinder.
\[ I = \frac{1}{2}MR^2 \]
Step 2: Express masses using density.
Mass \( \propto \) Volume.
For original cylinder: \[ M_1 \propto \pi R^2 L \] For carved cylinder: \[ M_2 \propto \pi \left(\frac{R}{3}\right)^2 \left(\frac{L}{2}\right) = \frac{\pi R^2 L}{18} \]
Step 3: Write moments of inertia.
\[ I_1 = \frac{1}{2} M_1 R^2 \] \[ I_2 = \frac{1}{2} M_2 \left(\frac{R}{3}\right)^2 \]
Step 4: Take ratio.
\[ \frac{I_1}{I_2} = \frac{M_1 R^2}{M_2 (R^2/9)} = 9 \times \frac{M_1}{M_2} \] \[ \frac{M_1}{M_2} = 18 \Rightarrow \frac{I_1}{I_2} = 9 \times 18 = 162 \]