Question

Statement 1: Net torque acting on a system can be given as
${\overrightarrow{\tau }}_{\text{net}}=\overrightarrow{r}\times \frac{d{\overrightarrow{P}}_{\text{sys}}}{dt}$
Where $\frac{d{\overrightarrow{P}}_{\text{sys}}}{dt}$ is the rate of change of momentum of the system.
Statement 2: Average torque acting on a system can be given as
${\overrightarrow{\tau }}_{\text{avg}}=\overrightarrow{r}\times \frac{\Delta {\overrightarrow{P}}_{\text{sys}}}{\Delta t}=\overrightarrow{r}\times \frac{({\overrightarrow{P}}_f-{\overrightarrow{P}}_i)}{\Delta t}$
where $\frac{\Delta {\overrightarrow{P}}_{\text{sys}}}{\Delta t}$ is change in momentum of system in time $\left(\Delta t\right)$

• A. Statement 1 is correct statement 2 is incorrect.
• B. Statement 1 is incorrect statement 2 is correct.
• C. Both statements are correct.
• D. Both statements are incorrect.

Answer 1

The correct option is C Both statements are correct.

We know,
Net external force acting on the system;
${\overrightarrow{F}}_{\text{net}}=\frac{d{\overrightarrow{P}}_{\text{sys}}}{dt}$
and
${\overrightarrow{\tau }}_{\text{net}}=\overrightarrow{r}\times {\overrightarrow{F}}_{\text{net}}$
$\Rightarrow {\overrightarrow{\tau }}_{\text{net}}=\overrightarrow{r}\times \frac{d{\overrightarrow{P}}_{\text{sys}}}{dt}$
Similarly,
Average external force acting on the system in time $\Delta t\text{sec}$;
$F_{\text{avg}}=\frac{\Delta {\overrightarrow{P}}_{\text{sys}}}{\Delta t}$
and
${\overrightarrow{\tau }}_{\text{avg}}=\overrightarrow{r}\times {\overrightarrow{F}}_{\text{avg}}$
$\Rightarrow {\overrightarrow{\tau }}_{\text{avg}}=\overrightarrow{r}\times \frac{\Delta {\overrightarrow{P}}_{\text{sys}}}{\Delta t}$
$=\overrightarrow{r}\times \frac{({\overrightarrow{P}}_{\text{final}}-{\overrightarrow{P}}_{\text{initial}})}{\Delta t}$
$\therefore$ Both statements are correct.