Establish the relation between torque and angular acceleration. Hence define the moment of inertia.

Torque and angular acceleration are related through the concept of rotational dynamics. Torque, denoted by the symbol "τ" (tau), is defined as the product of force and the perpendicular distance from the axis of rotation to the line of action of the force. It causes an object to rotate or change its angular motion. The formula for torque is:

τ = I α

where:

τ is the torque
I is the moment of inertia
α is the angular acceleration
Angular acceleration, denoted by "α," represents how quickly the angular velocity of an object changes over time. It is measured in radians per second squared (rad/s²). The angular acceleration depends on the torque applied and the moment of inertia of the object.

The moment of inertia, denoted by "I," is a measure of an object's resistance to changes in its rotational motion. It quantifies how mass is distributed around the axis of rotation. It depends on both the mass and the distribution of mass in the object. The moment of inertia can be thought of as the rotational analog of mass in linear motion.

The moment of inertia can be calculated using different formulas depending on the shape and distribution of mass in the object. For example, for a point mass rotating about a fixed axis, the moment of inertia is given by:

I = m r²

where:

I is the moment of inertia
m is the mass of the object
r is the perpendicular distance from the axis of rotation to the point mass
For more complex objects with distributed mass, such as a rotating solid cylinder or a rotating rod, the formulas for moment of inertia are different and involve integrals.

In summary, torque and angular acceleration are related through the equation τ = I α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. The moment of inertia, I, represents an object's resistance to changes in its rotational motion and depends on the mass and distribution of mass in the object.