When a rigid body rotates about an axis, it possesses kinetic energy. All the kinetic energy of a rotating body is rotational kinetic energy (Kr). Let us now find the kinetic energy of a rotating body. The particle of mass m1 follows a circular path of radius r1. The magnitude of the linear or tangential velocity of the particle on this circle is v₁.
Rotational kinetic energy of the particle = ½ m₁ v₁² = ½ m₁ (r₁ ω)² = ½ m₁ r₁² ω².
Similarly, the rotational kinetic energy of particles of masses m₂, m₃ …… are ½ m₂ r² ω², m₃ r² ω²… respectively.
The rotational kinetic energy Kr of the body s equal to the sum of the rotational kinetic energies of all the particles.
= ½ (m₁ r₁² + m₂ r₂² + m₃ r₃² + …..) ω²
= ½ (∑mᵢ rᵢ)² ω²But ∑mᵢ rᵢ² is the moment of inertia I of the body about the given axis of rotation.
Rotational K.E. of the body, Kr = I ω².
Thus the rotational K.E of a body is equal to half the product of the moment of inertia of the body and the square of the angular velocity of the body about the given axis of rotation, Note that once again we could have guessed its general form. In analogy to ½ m v², we see that v is replaced by ω and m by I.
For continuous mass distribution, the moment of inertia, I, is given by
I = ∫r² dm
Where r is the distance of the mass-element dm from the axis of rotation. The integration is carried out over the entire mass distribution.
Moment of inertia of a system of particles depends on
1. Axis of rotation
2. Mass of the system
3. Distribution of mass in the body
Moment of inertia plays same role in rotational motion as mass plays in translational motion. It is the property of the body by which body opposes any change in its state of rotational motion.