Moment of Inertia

Let C be the centre of mass of a rigid body. Suppose it rotates about an axis through this point.

K = ½mv2

Angular speed (ω) is related to linear speed (v) through the equation

v = rω

K = ½m(rω)2

K = ½mr2ω2

K = ½Iω2

I = mr2

The quantity I is called the moment of inertia of the body.

Moment of inertia is defined with reference to an axis of rotation. Therefore, whenever you mention moment of inertia, the axis of rotation must also be specified. In the present case, I is the moment of inertia about an axis passing through the centre of the square and normal to the plane.

The moment of inertia is expressed in kg m2. The moment of inertia of a rigid body is often written as

I = MK2

where M is the total mass of the body and K is called the radius of gyration of the body. The radius of gyration is that distance from the axis of rotation where the whole mass of the body can be assumed to be placed to get the same moment of inertia which the body actually has.

The moment of inertia of a body about an axis depends on the distribution of mass around that axis. If the distribution of mass changes, the moment of inertia will also change.

Physical Significance of Moment of Inertia

The physical significance of moment of inertia is that it performs the same role in rotational motion that the mass does in linear motion. 

Just as the mass of a body resists change in its state of linear motion, the moment of inertia resists a change in its rotational motion.

Equations of Motion for Uniformly Rotating Rigid Body

Consider an object rotating about an axis passing through O and normal to its plane. If it is rotating with a constant angular velocity ω, then it will turn through an angle θ in t seconds such that

θ = ωt

However, if the object is subjected to constant torque (turning effect of force), it will undergo a constant angular acceleration. The following equations describe its rotational motion:

ωf = ωi + αt

where ωi is initial angular velocity and ωf is final angular velocity.

Similarly,

θ = ωit + ½αt2

ωf2 = ωi2 + 2αθ

where θ is angular displacement in t seconds.