Question

Explain at least two ways that the moment of inertia can be doubled.

Answer 1

Moment of inertia can be defined w.r.t. rotation axis, as a quantity that decides the amount of torque required for a desired angular acceleration or a property of a body due to which it resists angular acceleration. The formula for moment of inertia is the “sum of the product of mass” of each particle with the “square of its distance from the axis of the rotation”. The formula of Moment of Inertia is expressed as

The moment of inertia depends on mass of the body, shape and size of the body and distribution of mass about the axis of rotation.

First Way: The moment of inertia of a body is directly proportional to its mass and increases as the mass is moved further from the axis of rotation.

As,

So, if we double the value of mass, the moment of inertia get doubled.

Second Way: By conservation of angular momentum L, the relation between moment of inertia I, and angular velocity , is

Therefore, if I increases, decreases so that the total angular momentum L is a constant. The vice versa is also true. If I decreses, increases.

As by the mathematical form:

Thus, when angular velocity becomes half, then moment of inertia get doubled.

Third Way: As, we know that:

here,

I = moment of inertia

= angular velocity

If the kinetic energy is constant, then,

such that,

Moment of Inertia is anti proportional to the squared of Angular Velocity.

So if the angular velocity becomes , then moment of inertia will be double.