Rotational kinetic energy and moment of inertia of a rigid body

Moment of inertia and its physical significance

According to Newton?s first law of motion, a body must continue in its state of rest or of uniform motion unless it is compelled by some external agency called force. The inability of a material body to change its state of rest or of uniform motion by itself is called inertia. Inertia is the fundamental property of the matter. For a given force, the greater the mass, the higher will be the opposition for motion, or larger the inertia. Thus, in translatory motion, the mass of the body measures the coefficient of inertia.

Similarly, in rotational motion also, a body, which is free to rotate about a given axis, opposes any change desired to be produced in its state. The measure of opposition will depend on the mass of the body and the distribution of mass about the axis of rotation. The coefficient of inertia in rotational motion is called the moment of inertia of the body about the given axis.

Moment of inertia plays the same role in rotational motion as that of mass in translatory motion. Also, to bring about a change in the state of rotation, torque has to be applied.

Rotational kinetic energy and moment of inertia of a rigid body

Consider a rigid body rotating with angular velocity ω about an axis XOX′. Consider the particles of masses m₁, m₂, m₃? situated at distances r_1, r ₂, r₃? respectively from the axis of rotation. The angular velocity of all the particles is same but the particles rotate with different linear velocities. Let the linear velocities of the particles be v₁,v₂,v₃ ? respectively.

Kinetic energy of the first particle = ?  m₁v₁²

But v₁=r₁ω

∴ Kinetic energy of the first particle

=1/2 . m₁(r₁ω)² =1/2 . m₂(r₂ω)²

Similarly,

Kinetic energy of second particle

=1/2 m²r₂²w²

Kinetic energy of third particle

= 1 / 2 .  m₃r₃²ω² and so on.

The kinetic energy of the rotating rigid body is equal to the sum of the kinetic energies of all the particles.

∴ Rotational kinetic energy

= 1 / 2 . ( m₁r₁²ω²  + m₂r₂²ω² + m₃r₃²ω² + ???. + m_nr_n²ω²)

= 1 / 2 . ω² ( m₁r₁²  + m₂r₂² + m₃r₃² + ???. + m_nr_n²)

In translatory motion, kinetic energy = 1 /2 mv²

Comparing with the above equation, the inertial role is played by the term ∑ m_nr_n². This is known as moment of inertia of the rotating rigid body about the axis of rotation. Therefore the moment of inertia is I = mass ? (distance )²

Kinetic energy of rotation = 1/2 ω²I

When ω = 1 rad s^-1, rotational kinetic energy

= ER = 1/2 (1)²I

 (or) I = 2E_R

It shows that moment of inertia of a body is equal to twice the kinetic energy of a rotating body whose angular velocity is one radian per second.

The unit for moment of inertia is kg m² and the dimensional formula is ML².

Radius of gyration

The moment of inertia of the rotating rigid body is,

I = ∑ m_ir_i² = m₁r₁² + m₁r₁² + ?? + m_nr_n²

If the particles of the rigid body are having same mass, then

m₁ = m₂ = m₃ =?.. = m (say)

∴ The above equation becomes,

I = nm[ r₁² + r₂² +?..+  r_n²]/n

where n is the number of particles in the rigid body.

∴ I = MK²

where M = nm, total mass of the body and K ² = [ r₁² + r₂² +?..+  r_n²]/n

Here K is called as the radius of gyration of the

rigid body about the axis of rotation.

The radius of gyration is equal to the root mean square distances of the particles from the axis of rotation of the body.

The radius of gyration can also be defined as the perpendicular distance between the axis of rotation and the point where the whole weight of the body is to be concentrated.

Also from the equation (2)  K² = r/M