Rotational Dynamics

Centre of Mass of a Rigid Body:

The centre of mass of a rigid body is a point whose position is fixed with the body as a whole. This point may or may not be within the body. It is possible that there may be no matter in centre of mass. The position of center of mass of a rigid body depends upon shape of the body , distribution of mass in the body. For a symmetrical body centre of mass is at the geometrical centre.

Centre of gravity and centre of mass:

Centre of gravity of a body is defined as the point where the whole weight of the body can be supposed to act.

The centre of mass of a body is the point where the whole mass of the body can be supposed to be concentrated.

Torque :

Torque is defined as product of magnitude of force and the lever arm of the force.

Angular momentum:

it is the product of linear momentum and lever arm of the momentum.

Moment of inertia:

Moment of inertia of a rigid body about a fixed axis is defined as the sum of the products of the masses of all the particles constituting the body and the squares of their respective distances from the axis of rotation.

Radius of gyration:

It is defined as the distance from the axis of rotation at which if whole mass of the body were concentrated, the moment of inertia would be same as the actual distribution of mass.

Factors on which moment of inertia depends:

• mass of the body
• position of the axis of rotation
• distribution of mass about the axis of rotation.

Theorem of perpendicular axis:

The moment of inertia of a plane lamina about axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes in its own plane intersecting each other at the point through which the perpendicular axis passes.

Theorem of parallel axis:

The moment of inertia of a body about any axis is equal to its moment of inertia about a parallel axis through its centre of gravity plus the product of mass of the body and the square of perpendicular distance between the two parallel axes.

Three laws of rotational motion:

• Unless an external torque is applied to it , a body in a state of rest or uniform rotational motion about its fixed axis of rotation remains unchanged.
• The rate of change of angular momentum of a body about a fixed axis of rotation is directly proportional to the torque applied and takes place in direction of the torque.
• When a torque is applied to by one body on another , an equal and opposite torque is applied by the latter on the former about the same axis of rotation.