Kinetic energy of ith particle is
K_i= ½ m_iv_i² ……(2)
It means that kinetic energy of whole body is sum of kinetic energies of i particles
Therefore, K = K_i = ∑ (½ m_i v_i²) ……(3)
Therefore, K_i = ½ ∑(m_i (ω r_i)²) since v_i = ω r_i
∴ K_i = ½ ∑ (m_i ω² r_i²)
as, ω² is constant
Therefore K_i = ½ (∑(m_i r_i²)) ω²
The term ∑(m_i r_i²) signifies moment of inertia denoted by I. i.e. I = ∑(m_i r_i²)
Therefore, K = ½ I ω² … (4)
Now compare equation (2) and equation (4), it an be seen that moment of inertia I is the analogue of mass in rotational motion.

Moment of inertia of continuous mass distribution

Consider small element of body. Let mass of element be dm and perpendicular distance from axis of rotation is r. Then the moment of inertia is calculated as,
I = ∫ r² dm

Significance of Moment of Inertia

• In rotational motion inertia of body resist the change in its state of rest or motion with a uniform angular velocity. Say that body is rotating with constant angular velocity. Change the rotational motion by applying torque in direction of motion or in opposite direction of motion. It body require large torque to change its shape then body posses greater inertia. This suggest that moment of inertia is greater about the axis of rotation.
• Mass is independent but moment of inertia is not a fixed quantity. However, it depends on
  1. Mass of the body
  2. Shape and size of the body
  3. Distribution of mass about the axis of rotation
  4. Position and orientation of axis of rotation

Unit and Dimension of Moment of Inertia

SI unit of moment of inertia is kg m², whereas, dimension is ML²

Keywords: Linear velocity, Angular speed, Kinetic energy, Mass, Inertia