State the law of conservation of angular momentum and explain it with the example of a girl sitting on a swivel chair.

(N/A) Law of Conservation of Angular Momentum: If the net external torque acting on a system is zero,the total angular momentum of the system remains constant.
Mathematical Derivation:
Angular momentum for rotational motion about a fixed axis is given by $\overrightarrow{L} = I \vec{\omega}$.
Differentiating with respect to time $t$ on both sides:
$\frac{d \overrightarrow{L}}{d t} = I \frac{d \vec{\omega}}{d t} = I \vec{\alpha} = \vec{\tau}$.
If the external torque $\vec{\tau} = 0$,then $\frac{d \overrightarrow{L}}{d t} = 0$,which implies $\overrightarrow{L} = \text{constant}$.
Illustration:
Consider a girl sitting on a rotating swivel chair with her arms outstretched. Her moment of inertia is $I_1$ and angular velocity is $\omega_1$. When she pulls her arms inward,her moment of inertia decreases to $I_2$ (where $I_2 < I_1$). Since the external torque is zero,the angular momentum is conserved: $I_1 \omega_1 = I_2 \omega_2$. Because $I_2 < I_1$,it follows that $\omega_2 > \omega_1$. Thus,the angular speed of the girl increases when she folds her arms.