Question

find the relation between torque and angular momentum.

Answer 1

$$\begin{gathered} \overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F} \\ =\overrightarrow{r}\times \frac{d\overrightarrow{p}}{dt} \\ =\overrightarrow{r}\times \left(\overrightarrow{\omega }\times \overrightarrow{p}\right) \\ =-\overrightarrow{p}\times (\overrightarrow{r}\times \overrightarrow{\omega })-\overrightarrow{\omega }\times (\overrightarrow{p}\times \overrightarrow{r}) \\ =\overrightarrow{p}\times (\overrightarrow{\omega }\times \overrightarrow{r})+\overrightarrow{\omega }\times (\overrightarrow{r}\times \overrightarrow{p}) \\ =\overrightarrow{p}\times \frac{d\overrightarrow{r}}{dt}+\overrightarrow{\omega }\times \overrightarrow{L} \\ =\overrightarrow{\omega }\times \overrightarrow{L} \\ where\frac{d\overrightarrow{r}}{dt}=\overrightarrow{\omega }\times \overrightarrow{r}and\frac{d\overrightarrow{p}}{dt}=\overrightarrow{\omega }\times \overrightarrow{p} \end{gathered}$$