A small mass m is attached to a massless string whose other...

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in figure. The mass is undergoing circular motion in $x-y$ plane with centre $O$ ? and constant angular speed $\omega$. If the angular momentum of the calculated about $O$ and Pand denoted by $\vec{L}_{O}$ and $\vec{L}_{P}$ respectively, then (a) $\vec{L}_{Q}$ and $\vec{L}_{p}$ do not vary with time. (b) $\vec{L}_{O}$ varies with time while $\vec{L}_{P}$ remains constant. (c) $\vec{L}_{O}$ remains constant while $\vec{L}_{P}$ varies with time. (d) $\vec{L}_{O}$ and $\vec{L}_{p}$ both vary with time.

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in figure. The mass is undergoing circular motion in $x-y$ plane with centre $O$ ? and constant angular speed $\omega$. If the angular momentum of the calculated about $O$ and Pand denoted by $\vec{L}_{O}$ and $\vec{L}_{P}$ respectively, then
(a) $\vec{L}_{Q}$ and $\vec{L}_{p}$ do not vary with time.
(b) $\vec{L}_{O}$ varies with time while $\vec{L}_{P}$ remains
constant.
(c) $\vec{L}_{O}$ remains constant while $\vec{L}_{P}$ varies with
time.
(d) $\vec{L}_{O}$ and $\vec{L}_{p}$ both vary with time.

Key Concepts

Central Force Motion

In the context of a particle moving under a central force—where the force is always directed towards or away from a fixed center—the angular momentum about that center remains constant because the force does not exert any torque about that point. This is an example of the general principle of conservation of angular momentum, which holds true when the net external torque is zero with respect to the chosen center.

Reference Point Dependence

The calculation of angular momentum is inherently dependent on the reference point or origin chosen. In circular motion, if the center of rotation is taken as the reference point, forces such as the centripetal force can produce zero torque about that point, leading to a constant angular momentum. However, when the reference point is offset from the center of the motion, the same forces may produce a non-zero torque, causing the angular momentum computed about that point to vary with time.

Angular Momentum

Angular momentum is a vector quantity defined as the cross product of the position vector and the linear momentum of a particle. This concept is pivotal in dynamics because its conservation or variation reveals the presence of torques. The definition and computation of angular momentum depends on the reference point, meaning that a change in the chosen origin can lead to differences in its calculated value, even for the same physical motion.

Torque and Angular Momentum

The time evolution of angular momentum is governed by the net external torque acting on the system, as expressed in the equation dL/dt = ?. When the net torque about a chosen point is zero, the angular momentum calculated about that point is conserved. Conversely, if there is a nonzero torque (even if the magnitude of the angular velocity is constant), the angular momentum about that point can vary with time.

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