An overview

The angular acceleration of any system of particles is defined as the sum total of the product of every particle mass and square of distance of each of the particles from the axis of rotation of the entire body.

The physical quantity which resists the angular acceleration of the body is known as Moment of Inertia (MOI). MOI in terms of torque is defined as the external torque needed to produce the angular acceleration around the axis of rotation of the body (system of particles). It is also termed rotational inertia as it resists angular acceleration, basically it is a rotational analog of the ‘mass’ of the object in linear motion. Hence, the concept of point mass becomes the fundamental basis of the moment of inertia of any object (system of particles).

Mathematically, MOI = m_ir_i², where m_i is the mass of the ith particle and r_i² is the square of the distance of the ith particle from the axis of rotation.

• SI unit of MOI is kgm²
• Dimensional formula I [M¹L²T⁰]

The axis of rotation plays a significant role in the calculation of MOI, hence the selection of axis greatly impacts the value of MOI, i.e. with change in axis, significant variation in MOI is observed.

The role of MOI in rotational motion is the same as the role of mass in linear motion. As the mass is the measure of inertia of the body, similarly MOI is the measure of the resistance to bring about a change in the rotational motion.

Factors affecting Moment of Inertia

Following are the factors that affect MOI

• Dimensions of the object
• Density with which the particles are packed in the object
• Axis of rotation and distribution of mass around it

We can further categorize rotating body systems as follows:

1. Discrete (System of particles).
2. Continuous (Rigid body).

For system of particles with discrete distribution the moment of inertia of system of particles can be calculated as:

MOI= m_ir_i²

where m_i is the mass of the ith particle and r_i² is the square of the distance of the ith particle from the axis of rotation.

For a system of particles with continuous mass distribution, the moment of inertia can be calculated by using the method of integration. Consider an infinitesimal particle of small mass ‘dm’ and positioned at a distance ‘x’ from the axis of rotation. The moment of inertia can be calculated as

I = ∫ r² dm

The objects, the shape of which cannot be described mathematically, the moment of inertia can be calculated experimentally. One such experiment which describes the relationship between the time period of oscillation of a torsion pendulum and the mass of the suspended bob. The time period of the oscillation depends upon the stiffness of the wire with which the mass is hanged. More is the MOI of the object, longer will be the time period.

Application of Moment of Inertia

• The heavy mass mounted on the shaft of an engine is known as flywheel, the high magnitude of the moment of inertia of the flywheel helps in storing a large amount of energy
• If we consider the hollow and solid shaft of the same mass, the power transmitted by the hollow shaft will be more than the solid shaft, which  is because the moment of inertia of the hollow shaft is more than the solid shaft
• The construction of a ship is actually an application of moment of inertia, the shipbuilding is done in such a way that the ship will never sink while pitching, but may sink by rolling in water

Standard formulae to calculate Moment of Inertia for different Objects