Equation of Motion of a Rotating Rigid body

Equation of Motion of a Rotating Rigid body:

Consider a rigid body rotating with the angular velocity ‘ω’ with respect to a fixed axis. At that time the torque acting on the rigid body is given by the rate of change of angular momentum. i.e.

Torque acting on the body,\begin{align*}\overrightarrow{\tau}=\frac{\mathrm{d} \overrightarrow{L}}{\mathrm{d} x}\end{align*}

Where, \(\begin{align*}\overrightarrow{L}\end{align*}\)is the angular momentum of the rigid body.

Since we have,

\begin{align*}\overrightarrow{L}=I\overrightarrow{\omega }\end{align*}, I= M.I of the rigid body.

Then, above equation becomes,

\begin{align*}\tau =\frac{\mathrm{d} (I\overrightarrow{\omega })}{\mathrm{d} x}\end{align*}\begin{align*}\tau =\frac{I\mathrm{d} (\overrightarrow{\omega })}{\mathrm{d} x}\end{align*}\begin{align*}\therefore \tau =I\alpha\end{align*}

Where, \(\begin{align*}\alpha =\frac{\mathrm{d} \overrightarrow{\omega }}{\mathrm{d} x}\end{align*}\) is the angular acceleration of the rigid body.

References:

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014