Moment of Inertia

The moment of inertia is a physical quantity which describes how easily a body can be rotated about a given axis. It is a rotational analogue of mass, which describes an object's resistance to translational motion. Inertia is the property of matter which resists change in its state of motion. Inertia is a measure of the force that keeps a stationary object stationary, or a moving object moving at its current speed. The larger the inertia, the greater the force that is required to bring some change in its velocity in a given amount of time. Suppose a heavy truck and a light car are both at rest, then intuitively we know that more force will be required to push the truck to a certain speed in a given amount of time than will be needed to push the car to that same speed in the same amount of time.

Similarly, moment of inertia is that property where matter resists change in its state of rotatory motion. The larger the moment of inertia, the greater the amount of torque that will be required to bring the same change in its angular velocity in a given amount of time. Here, torque and angular velocity are the angular analogues of force and velocity, relating to moment of inertia in the exact same way that force and velocity relate to mass.

The moment of inertia of an object made up of 𝑛 point particles about an axis is given as follows:

where 𝑚𝑖 is the mass of one of the point particles making up the object and 𝑟𝑖 is the distance from that point particle to the axis (measured perpendicular to the axis, which is the same as the minimum distance from the point particle to the axis). This means that the moment of inertia is higher for objects where the mass is distributed a long way away from the axis of rotation.