Question

Define torque. Derive the relation between angular momentum and torque.

Answer 1

Dear Student,

Torque (τ): It is an moment of force, i.e. tendency of a force to rotate an object about an axis.

τ = r × f
where r is the perpendicular distance from the axis of the rotation and f is the applied force.

$$\begin{gathered} \mathrm{For}\mathrm{rotational}\mathrm{motion},\mathrm{torque}\mathrm{is}\mathrm{given}\mathrm{by}: \\ \tau =I\alpha ......\left(1\right) \\ \mathrm{where},I=\mathrm{moment}\mathrm{of}\mathrm{inertia} \\ \alpha =\mathrm{angular}\mathrm{acceleration} \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{for}\mathrm{rotational}\mathrm{body},\mathrm{angular}\mathrm{momentum}\mathrm{is}\mathrm{conserved}. \\ L=I\omega .....\left(2\right) \\ \mathrm{where},\omega =\mathrm{angular}\mathrm{velocity} \\ \mathrm{Also}, \\ \alpha =\frac{d\omega }{dt}......\left(3\right) \\ \mathrm{taking}\mathrm{derivation}\mathrm{of}\mathrm{equation}\left(2\right)\mathrm{w}.\mathrm{r}.\mathrm{t}\mathrm{time}, \\ \frac{dL}{dt}=I\frac{d\omega }{dt}......\left(4\right) \\ \mathrm{from}\left(3\right)\&\left(4\right), \\ \frac{dL}{dt}=I\alpha .....\left(5\right) \\ \mathrm{from}\mathrm{equation}\left(1\right)\&\left(5\right) \\ \tau =\frac{dL}{dt} \end{gathered}$$