Question

A ring has greater moment of inertia than a circular disc of same mass and radius, about an axis passing through its centre of mass perpendicular to its plane, because

Answer 1

Hint: The moment of inertia of a rigid body, also known as mass moment of inertia, angular mass, second moment of mass, or, more precisely, rotational inertia, is a quantity that determines the torque required for a desired angular acceleration about a rotational axis, in the same way that mass determines the force required for a desired acceleration. It relies on the mass distribution of the body and the axis selected, with greater moments necessitating more torque to affect the rate of rotation.

Complete step by step solution:
It is an extended (additive) property: the moment of inertia for a point mass is simply the mass times the square of the perpendicular distance to the rotation axis. A rigid composite system's moment of inertia is equal to the sum of the moments of inertia of its component subsystems (all taken about the same axis). The second moment of mass with regard to distance from an axis is the simplest definition.
Because its whole mass is concentrated at the rim at the greatest distance from the axis, a ring has a greater moment of inertia.
When we talk about the moment of inertia of a disc, we may state that it is quite comparable to the moment of inertia of a solid cylinder of any length. However, we must treat it as a special character when dealing with a disc. It's commonly used as a starting point for calculating the moment of inertia for various forms like cylinders and spheres.
The axis of rotation for a sphere is the disk's centre axis. It's written as: \[\dfrac{1}{2}M{{R}^{2}}\]
The axis of a ring will be in the centre. It's written like this: \[\frac{1}{2}M\text{ }({{a}^{2}}~+\text{ }{{b}^{2}})\]


Note: The rigid body is called an asymmetric top when all primary moments of inertia are distinct, the principal axes through the centre of mass are uniquely defined, and all principal moments of inertia are distinct. When two main moments are the same, the rigid body is termed a symmetric top, and the two corresponding principal axes have no unique option. The rigid body is termed a spherical top (though it does not have to be spherical) if all three primary moments are the same, and any axis can be regarded a principal axis, meaning that the moment of inertia is the same around any axis.