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Dynamics of Rotational Motion about a Fixed Axis

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Dynamics of Rotational Motion about a Fixed Axis - Lesson Summary

Using the analogy of translational motion and rotational motion, the moment of inertia is the rotational analogue of mass in translational motion. Similarly, torque in rotational motion plays the same role as that of force in linear motion. The expressions for a rotational body about a fixed axis keeping in mind the dynamics of the system are derived. A rigid body rotating about a fixed axis is considered. As the axis is fixed, only the components of torque, which are along the axis of rotation, can cause the body to rotate about the axis.
 
We know that torque τ = r x F. Thus, the torque acting along the axis of rotation must be perpendicular to the plane containing the position vector r and force F.  To calculate torque acting on such a rigid body, we will consider only those forces which lie in a plane perpendicular to the axis of rotation so that the torque produced by them will lie along the axis of rotation.  Only the components of position vectors that lie in a plane perpendicular to the axis of rotation need to be considered for the torque along the axis of rotation.
 
Let’s now try to find the work done by the torque for a rigid body rotating about a fixed axis keeping these considerations in mind. Consider a particle in a rigid body located at P with its position vector represented by r. The rigid body is rotating about the z-axis.
 

 
When work is done on a rotating body, its energy must increase. However, in the case of a rigid body, the change in its potential energy must be zero as there is no change in intermolecular distances. Hence the increase in energy when work is done on a rotating rigid body increases the kinetic energy of the body.  Therefore, the rate of work done on a rigid body rotating about a fixed axis must be equal to the rate of increase in its rotational kinetic energy. The rate of work done or power is equal to the rate of change of kinetic energy.

Newton’s Second Law of Rotational Motion states that the angular acceleration of a rotating body is directly proportional to the applied torque and is inversely proportional to the moment of inertia of the body.
 

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