Question

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the x-y plane with center at O and constant angular speed $\omega$. If the angular momentum of the system calculated about O and P are denoted by ${\overrightarrow{L}}_{0}and{\overrightarrow{L}}_r$ respectively, then

• A. ${\overrightarrow{L}}_0$ and ${\overrightarrow{L}}_r$ do not vary with time
• B. ${\overrightarrow{L}}_0$ varies with time while ${\overrightarrow{L}}_r$ remains constant
• C. ${\overrightarrow{L}}_0$ remains constant while ${\overrightarrow{L}}_r$ varies with time
• D. ${\overrightarrow{L}}_0$ and ${\overrightarrow{L}}_r$ both vary with time

Answer 1

The correct option is C

${\overrightarrow{L}}_0$ remains constant while ${\overrightarrow{L}}_r$ varies with time

${\overrightarrow{L}}_0=m(\overrightarrow{r}\times {\overrightarrow{r}}_0)\times \overrightarrow{v}=mr{\overrightarrow{e}}_r\times mr{\overrightarrow{e}}_t=m\omega r^2\hat{z}\overrightarrow{L}=m(\overrightarrow{r}\times {\overrightarrow{r}}_e)\times \overrightarrow{v}=m(r{\overrightarrow{e}}_r-h\hat{z})\times mr{\overrightarrow{e}}_t=mh\omega r{\overrightarrow{e}}_r+m\omega r^2\hat{z}$