question_answer

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

Answer:

                Power in rotational motion, \[P=\tau \omega \] Rotational K.E., \[K=\frac{1}{2}I{{\omega }^{2}}\] Work done in rotational motion, \[W=\]Energy stored as rotational K.E.  \[=\frac{1}{2}I{{\omega }^{2}}\] \[\therefore \] \[P=\frac{dW}{dt}=\frac{d}{dt}\left( \frac{1}{2}I{{\omega }^{2}} \right)=\frac{1}{2}.I\times 2\omega \frac{d\omega }{dt}\] or            \[\tau \omega =I\omega \alpha \]           \[\left[ \because \frac{d\omega }{dt}=\alpha  \right]\] \[\therefore \]  \[\tau =I\alpha .\]