Question

The moment of inertia of a solid sphere about an axis passing through its centre is 0.8kgm^2. The moment of inertia of another solid sphere whose mass is same as mass of first sphere, but the density is 8 times density of first sphere, about an axis passing through its centre is

Answer 1

Given :

The moment of inertia of solid sphere = 0.8 kgm²

Density of first sphere = ρ

Density of another sphere = 8ρ

To Find :

The moment of inertia of another sphere

Solution :

• Moment of inertia ∝ Radius²

             I ∝ R²

             I₁/I₂ = R₁²/R₂²

• The density of a body  ∝ 1/Radius³

            ρ ∝  1 /R³

            ρ₁/ρ₂  = R₂³/R₁³

            R₁/R₂ =   (ρ₂/ρ₁)^(1/3)

• By substituting in above relation we get

         I₁/I₂  =  (ρ₂/ρ₁))^(2/3)

         I₁/I₂  = 4

         I₂ = I₁/4

         I₂ = 0.2kgm²

The moment of inertia of second sphere is 0.2kgm²

Answer 2

Answer: The correct answer is 0.2 $kgm^2$

Explanation:

Let the radius of first sphere be $R_1$ and radius of second sphere be $R_2$.

Given in the question,

$I_{cm_1}$ = 0.8 $kgm^2$ (Moment of Inertia about an axis passing through its centre)

mass of both sphere are same = m

For a solid sphere we know that density  ρ α $\frac{1}{R^3}$

Let ρ1 be the density of first sphere and ρ2  be the density of second sphere.

∴ ρ2 = 8ρ1

∴ ρ1 / ρ2 =  ($\frac{R_2}{R_1}$)$.^3$   ................. equation 1

$I_{cm_2}$ = ? (Moment of Inertia about an axis passing through its centre)

∴Moment of Inertia of a sphere =  $I_{sphere}$ = $\frac{2}{5}$ X $MR^2$

Since, the mass is same, we can rewrite above as,

$I_{sphere}$  α $R^2$  ( Directly proportional)

∴ $\frac{I_1}{I_2}$ = ($\frac{R_1}{R_2}$)$.^2$      ................. equqation 2

From equations 1 and 2 we can write,

$\frac{I_1}{I_2}$ =  (ρ2 / ρ1)$.^\frac{2}{3}$

but ρ2 = 8ρ1

∴ $\frac{I_1}{I_2}$ =  (8ρ1 / ρ1)$.^\frac{2}{3}$

∴  $\frac{I_1}{I_2}$  = (8)$.^\frac{2}{3}$

and $I_1$ = 0.8 (GIven in the question)

$I_2$ = $\frac{0.8}{4}$

$I_2$ = 0.2 $kgm^2$

∴ Moment of Inertia of second sphere about an axis passing through its centre is 0.2 $kgm^2$.