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 ___

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

Q. 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 ____.

Correct Answer: 162

Explanation

Moment of inertia of a solid cylinder about its own axis is given by:

I = ½ M R²

Let density of the material be ρ.

Mass of original cylinder:

M₁ = ρ × πR² × L

Moment of inertia of original cylinder:

I₁ = ½ (ρπR²L) R² I₁ = ½ ρπ L R⁴

Now consider the smaller co-centric cylinder.

Its radius = R/3 Its length = L/2

Mass of smaller cylinder:

M₂ = ρ × π (R/3)² × (L/2)

M₂ = ρπR²L / 18

Moment of inertia of smaller cylinder:

I₂ = ½ M₂ (R/3)²

I₂ = ½ × (ρπR²L / 18) × (R² / 9)

I₂ = ρπ L R⁴ / 324

Now take the ratio:

I₁ / I₂ = (½ ρπ L R⁴) ÷ (ρπ L R⁴ / 324)

I₁ / I₂ = ½ × 324

I₁ / I₂ = 162

Hence, the required ratio is 162.

Explanation

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