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 ti…
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.
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Q1: What happens to angular momentum when net torque is zero?
When the net external torque acting on a system is zero, the system's total angular momentum remains constant. This principle applies even though individual particles within the system may experience internal forces and torques. Newton's third law ensures these internal torques cancel each other out, leaving angular momentum conserved.
Q2: How does moment of inertia affect angular velocity in a rotating system?
Angular momentum equals moment of inertia multiplied by angular velocity. When moment of inertia increases while angular momentum stays constant, angular velocity must decrease proportionally. For example, placing a box on a rotating merry-go-round doubles its moment of inertia, causing angular velocity to decrease to half its initial value.
Q3: Why can ice skaters spin faster when they pull their arms inward?
Pulling arms inward decreases moment of inertia. Since net torque from friction is negligible, angular momentum remains constant. To conserve angular momentum with reduced moment of inertia, rotational speed must increase. The work required to pull arms inward also increases rotational kinetic energy while angular momentum stays constant.
Q4: How is rotational kinetic energy related to angular momentum?
Rotational kinetic energy equals half the moment of inertia times angular velocity squared. By substituting angular momentum for the product of moment of inertia and angular velocity, kinetic energy can be expressed directly in terms of angular momentum. This relationship shows how energy and momentum are interconnected in rotating systems.
Q5: What role does torque play in changing angular momentum?
The change of angular momentum with time equals the net torque acting on the system. This relationship, derived from rotational dynamics, means torque is the rotational equivalent of force. Only external torques change a system's total angular momentum; internal torques between particles cancel due to Newton's third law.
Q6: Why does an ice skater's kinetic energy increase when spinning faster?
When an ice skater pulls their arms inward, they perform work against the centrifugal effect. This work increases rotational kinetic energy even though angular momentum remains constant. In a frictionless environment, no energy escapes the system, so all work done goes into increasing spin rate and kinetic energy.
Q7: How do internal forces affect a system's total angular momentum?
Internal forces between particles produce internal torques, but Newton's third law ensures these torques are equal and opposite, canceling each other out. Therefore, internal forces cannot change a system's total angular momentum. Only net external torque can alter the total angular momentum of a system.