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): 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.
A uniform circular disc of radius \( R \) and mass \( M \) is rotating about an axis perpendicular to its plane and passing through its center. A small circular part of radius \( R/2 \) is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.
Rotational motion can be defined as the motion of an object around a circular path, in a fixed orbit.
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: