Physics Question on rotational motion

Question

The ratio of the radii of two solid spheres of same mass in 2:3. The ratio of the moments of inertia of the spheres about their diameters is:

Answer 1

The correct option is: (A): 4:9.

The moment of inertia of a solid sphere about its diameter (I) is proportional to its mass (m) and the square of its radius (r):

I ∝ m * r²

Given that the masses of both spheres are the same, we can set up a relationship between the radii of the spheres using the given ratio:

r₁ : r₂ = 2 : 3

Let's assume the common mass is 'm', and the radii of the spheres are 2r and 3r, respectively.

The moments of inertia of the two spheres are then:

I₁ = m * (2r)² = 4 * m * r² I₂ = m * (3r)² = 9 * m * r²

The ratio of the moments of inertia (I₁ : I₂) is:

I₁ : I₂ = 4 * m * r² : 9 * m * r² = 4 : 9

So, the ratio of the moments of inertia of the spheres about their diameters is indeed 4:9.