Question

A particle undergoes uniform circular motion. about which point on the plane of the circle, will the angular momentum of the particle remain conserved?

• A. inside the circle
• B. outside the circle
• C. center of the circle
• D. on the circumference of the circle.

Answer 1

The correct option is C

center of the circle

Explanation of the correct option:

In the case of option C.

1. Mathematically, $\overrightarrow{\tau }=\overrightarrow{r}\times {\overrightarrow{F}}_C$, where $\overrightarrow{\tau }=$torque, $\overrightarrow{r}=$radius, $\overrightarrow{F_C}=$centripetal force.
2. Now, $\overrightarrow{\tau }=r.f.\sin \theta =r.f.\sin \left(180\right)=0$
3. So, $\overrightarrow{\tau }=\frac{d\overrightarrow{l}}{dt}=0,or\overrightarrow{l}=$constant, where $\overrightarrow{l}=$angular momentum,
4. Hence, angular momentum is conserved.
5. In uniform circular motion, the only force acting on the particle is centripetal force. (which acts towards the centre).
6. The torque of this force about the centre is zero.
7. Hence, angular momentum about the centre remains conserved.
8. Thus, option C is the correct option.

Explanation of the incorrect options:

In the case of option A.

1. Here, by taking any point inside the circle, $\overrightarrow{r}$ will make an angle $\theta$ with the ${\overrightarrow{F}}_C$.
2. Thus, in this case, $\overrightarrow{\tau }=r.f.\sin \theta \ne 0$. hence momentum is not conserved in this case.
3. Thus, option A is an incorrect option.

In the case of option B.

1. Here, by taking any point outside the circle, $\overrightarrow{r}$ will make an angle $\theta$ with the ${\overrightarrow{F}}_C$.
2. Thus, in this case, $\overrightarrow{\tau }=r.f.\sin \theta \ne 0$. hence momentum is not conserved in this case.
3. Thus, option B is an incorrect option.

In the case of option D.

1. In this case, torque $\overrightarrow{\tau }=r.f.\sin 90\ne 0$, hence momentum is not conserved in this case.
2. Thus, option D is an incorrect option.

Hence, the correct option is option C.