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:
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:
4:9
2:3
8:27
17:21
The Correct Option is A
Solution and Explanation
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.
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Concepts Used:
Rotational Motion
Rotational motion can be defined as the motion of an object around a circular path, in a fixed orbit.
Rotational Motion Examples:
The wheel or rotor of a motor, which appears in rotation motion problems, is a common example of the rotational motion of a rigid body.
Other examples:
- Moving by Bus
- Sailing of Boat
- Dog walking
- A person shaking the plant.
- A stone falls straight at the surface of the earth.
- Movement of a coin over a carrom board
Types of Motion involving Rotation:
- Rotation about a fixed axis (Pure rotation)
- Rotation about an axis of rotation (Combined translational and rotational motion)
- Rotation about an axis in the rotation (rotating axis)