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: 

• A. 4:9
• B. 2:3
• C. 8:27
• D. 17:21

Answer 1

The Correct Option is A

To solve the problem, we need to determine the ratio of the moments of inertia of two solid spheres having the same mass but different radii in the ratio 2:3.

1. Moment of Inertia of a Solid Sphere:
The moment of inertia $I$ of a solid sphere about its diameter is given by the formula:
$ I = \frac{2}{5} M R^2 $
where:
$M$ is the mass of the sphere,
$R$ is the radius of the sphere.

2. Given Condition:
- Let the radii of the two spheres be $R_1$ and $R_2$, such that:
$ R_1 : R_2 = 2 : 3 $
- The masses of both spheres are the same.

3. Applying the Moment of Inertia Formula:
Since mass is constant, the ratio of their moments of inertia is directly proportional to the square of their radii:
$ \frac{I_1}{I_2} = \frac{R_1^2}{R_2^2} = \left( \frac{2}{3} \right)^2 = \frac{4}{9} $

Final Answer:
The ratio of the moments of inertia of the spheres about their diameters is 4 : 9.

Answer 2

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.