We know that rate of change of angular momentum is equal to the torque applied i.e.,
\(\frac{dL}{dt} = \tau\)
If external torque is zero, i.e., \(\tau\) = 0
the \(\frac{dL}{dt} = 0\)
or L = constant
or \(L = I\omega = constant\)
i.e., "In absence of external torque the angular momentum of a body remains unchanged or constant." This is the principle of conservation of angular momentum.
\(\because\ I\omega = constant\)
Therefore on decreasing the value of I, angular velocity increases and vice-versa.
\(\therefore\ I_1\omega_1 = I_2\omega_2\)
Example:
When a person rotates sitting on a rotating stool, then as he opens his hands away from his body his angular velocity decreases and as he starts closing his hands again, his angular velocity of rotation goes on increasing. Above fact can be explained on the basis of law of conservation of angular momentum according to which,
\(I\omega = constant\)
s the person opens his hands his moment of inertia increases, hence angular velocity ω decreases and hence vice-versa.
