question_answer

Using expressions for power and kinetic energy of rotational motion, derive the relation \[\tau =I\alpha \], Where letters have their usual meaning.

Answer:

                We know that power in rotational motion, \[P=\tau \omega \]                                                  ?.. (i) And K.E. of rotation, \[E=\frac{1}{2}I{{\omega }^{2}}\] As power = time rate of doing work in rotational motion, and work is stored in the body in the form of K.E. \[\therefore \]  \[P=\frac{d}{dt}\left( K.E.\,of\,rotaion \right)\] \[=\frac{d}{dt}\left( \frac{1}{2}I{{\omega }^{2}} \right)=\frac{1}{2}I\times 2\omega \left( \frac{d\omega }{dt} \right)\] \[P=I\omega \alpha \] Using (i), \[P=\tau \omega =\,I\omega \,\alpha \] or \[\tau =\,I\,\alpha \], which is the required relation.