11.10:
Conservation of Angular Momentum
A system rotating with angular velocity ω has total angular momentum equal to its moment of inertia times the angular velocity.
The change of angular momentum with time gives the net torque acting on the system. If the net torque acting on the system is zero, then the system's angular momentum is conserved.
The rotational kinetic energy of the system is expressed as half of the moment of inertia multiplied by the square of the angular velocity.
The rotational kinetic energy can also be expressed in terms of the angular momentum by substituting the product of the moment of inertia and the angular velocity as the system's total angular momentum.
Consider a merry-go-round in the park, rotating with an initial angular velocity and moment of inertia, respectively. Now, if a box is placed vertically on the merry-go-round, its moment of inertia is doubled. To keep the angular momentum conserved, its final angular velocity decreases by one-half of the initial angular velocity.
11.10:
Conservation of Angular Momentum
A system's total angular momentum remains constant if the net external torque acting on the system is zero. Considering a system that consists of n tiny particles, the angular momentum of any tiny particle may change, but the system's total angular momentum would remain constant. The principle of conservation of angular momentum only considers the net external torque acting on the system. While there are internal forces exerted by different particles within the system that also produce internal torques, Newton's third law of motion states that these torques are equal and opposite in nature, and cancel each other out.
As an example of conservation of angular momentum, consider ice skaters executing a spin. The net torque acting on them is very close to zero because there is relatively little friction between the skates and the ice. Also, the friction is exerted very close to the pivot point. Both the force and its the distance from the lever arm are small, so the torque is negligible. Consequently, the ice skaters can spin for a long time. They can also increase their rate of spin by pulling their arms and legs in. When they pull their arms and legs in, it decreases their moment of inertia; thus to keep the angular momentum constant, their rate of spin increases.
It is also interesting to note that their final rotational kinetic energy increases as their moment of inertia begins to decrease. The source of this additional rotational kinetic energy is the work required to pull the arms inward. Note that the skater's arms do not move in a perfect circle—they spiral inward. This work causes an increase in the rotational kinetic energy while their angular momentum remains constant. Since they are in a frictionless environment, no energy escapes the system. Thus, if they were to extend their arms to their original positions, they would rotate at the initial angular velocity, and their kinetic energy would return to its initial value.
This text is adapted from Openstax, University Physics Volume 1, Section 11.3: Conservation of Angular Momentum.