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 centre at $O$ and constant angular speed $ω$ . If the angular momentum of the system, calculated about $O$ and $P$ are denoted by ${\to L}_{O}$ and ${\to L}_{P}$ respectively, then

{\to L}_{O}

and

{\to L}_{P}

do not vary with time.

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{\to L}_{O}

varies with time while

{\to L}_{P}

remains constant.

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{\to L}_{O}

remains constant while

{\to L}_{P}

varies with time.

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{\to L}_{O}

and

{\to L}_{P}

both vary with time.

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Solution

The correct option is C ${\to L}_{O}$ remains constant while ${\to L}_{P}$ varies with time.
Angular momentim about $O$ :
As we know, angular momentum $\to L = m (\to r \times \to V)$
Here $\to r = - - \to O A$ . Since mass is undergoing circular motion, $\to V$ will be in the tangential direction of a circle in the plane of circular motion.

${\to L}_{O} = m (- - \to O A \times \to V)$
Angular momentum will be in z-direction (perpendicular to the plane of circular motion).
After some time, mass is at $B$ . Velocity will be in tangential direction. Hence
${\to L}_{O} = m (- - \to O B \times \to V)$ has the same magnitude and direction.
i.e ${\to L}_{O}$ remains constant.

Angular momentum about point $P$ :
Here, $\to r = - - \to P A$
and angular momentum will be $\to L = m (- - \to P A \times \to V)$
$\to V$ will be in tangential direction in the circular plane. Hence, the direction of angular momentum will be as shown in figure. (by right hand thumb rule).

After time $t$ , when particle is at $B$ , $\to r = - - \to P B$ and velocity vector will be in tangential direction.
So, angular momentum $\to L = m (- - \to O B \times \to V)$ and the direction of angular momentum (from right hand rule ) as shown in the figure.

Hence, magnitude of angular momentum will be same about point $P$ , but direction is changing.
$∴$ Angular momentum about point $O$ is constant, but about point $P$ it is varying.

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Similar questions

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 centre at O and constant angular speed $ω$ . If the angular momentum of the system, calculated about O and P are denoted by $_{O}$ and $_{P}$ respectively, then

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 $ω$ . If the angular momentum of the system calculated about O and P are denoted by ${\to L}_{0} a n d {\to L}_{r}$ respectively, then