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

Given:
A uniform solid cylinder of length L and radius R has moment of inertia I₁ about its axis.
A small concentric cylinder of length L/2 and radius R/3 is carved out and has moment of inertia I₂.

Step 1: Moment of inertia of a solid cylinder about its axis
I = (1/2) M R²

Step 2: Find mass of the large cylinder
Mass ∝ Volume
M₁ ∝ πR²L

So,
I₁ = (1/2) M₁ R²

Step 3: Find mass of the small cylinder
Radius = R/3
Length = L/2

M₂ ∝ π (R/3)² (L/2)
M₂ ∝ (πR²L) / 18

So,
M₂ = M₁ / 18

Step 4: Moment of inertia of small cylinder
I₂ = (1/2) M₂ (R/3)²
I₂ = (1/2) × (M₁ / 18) × (R² / 9)
I₂ = M₁ R² / 324

Step 5: Take ratio I₁ / I₂
I₁ / I₂ = [(1/2) M₁ R²] / [M₁ R² / 324]
I₁ / I₂ = 324 / 2
I₁ / I₂ = 162

Final Answer:
I₁ / I₂ = 162