Question

State and prove principle of conservation of angular momentum.

Answer 1

Hint -The law of conservation of Angular Momentum states that "When the net external torque acting on a system about a given axis is zero, the total angular momentum of the system about that axis remains constant."

Complete step-by-step answer:
The external torque on a body is zero, if the angular momentum of the body is conserved.

Let a particle of mass m whose position vector with respect to origin at any instant is

The linear velocity of the particle is given by \[\overrightarrow v = \dfrac{{\overrightarrow {dr} }}{{dt}}\]

The linear momentum and angular momentum of the body is given by $\overrightarrow p = m\overrightarrow v $ and $\overrightarrow l = \overrightarrow r \times \overrightarrow p $ about an axis through the origin.

The angular momentum $\overrightarrow l $ may change with time due to a torque on the particle.

\[
  \dfrac{{\overrightarrow {dl} }}{{dt}} = \dfrac{d}{{dt}}\left( {\overrightarrow r \times \overrightarrow p } \right) \\
   = \dfrac{{d\overrightarrow r }}{{dt}} \times \overrightarrow p + \overrightarrow r \times \dfrac{{d\overrightarrow p }}{{dt}} \\
   = \overrightarrow v \times \overrightarrow {mv} + \overrightarrow r + \overrightarrow F \\
   = \overrightarrow r \times \overrightarrow F \,\,\,\left( {\overrightarrow v \times \overrightarrow v = 0} \right) \\
   = \overrightarrow \tau \\
\]

Where \[\dfrac{{d\overrightarrow p }}{{dt}} = \overrightarrow F \] the force on the particle
Hence if $\overrightarrow \tau = 0,\,\dfrac{{\overrightarrow {dl} }}{{dt}} = 0$

$\therefore \overrightarrow l $ = constant, i.e. $\overrightarrow l $ is conserved.

Additional Information- Angular momentum is just linear momentum that is caused by any inclination to turn only so that an object stays the same distance away from a central point. Angular momentum is maintained as it preserves linear momentum. In a circular motion the distance that something moves is in terms of angle.

Note- The angular momentum is dependent on the object's rotational velocity but also on its rotational inertia if an object changes shape (rotational inertia). its angular velocity will also change if there is no external torque. It is the solar system whose angular momentum is conserved, but because the earth's mass is negligible relative to the sun, it can be assumed that the angular momentum of the planet is retained (if it is viewed as part of an independent system).