AP Physics 1- 6.4 Conservation of Angular Momentum- Study Notes- New Syllabus
AP Physics 1-6.4 Conservation of Angular Momentum – Study Notes
AP Physics 1-6.4 Conservation of Angular Momentum – Study Notes -AP Physics 1 – per latest Syllabus.
Key Concepts:
- Conservation of Angular Momentum
- System Selection & Angular Momentum
Conservation of Angular Momentum
Angular momentum of a system remains constant if the net external torque on the system is zero. This principle applies to rigid and non-rigid systems alike.
Main Points:
- Law of Conservation: If no external torque acts on a system, then the total angular momentum of the system remains conserved: $ L_i = L_f $
- System Composition: The total angular momentum of a system about a chosen axis is the sum of the angular momenta of its constituent parts about that axis.
- Cause of Change: Any change to a system’s angular momentum must be due to an interaction with its surroundings (i.e., a net external torque).
- Newton’s Third Law (Rotational Form): The angular impulse exerted by one object/system on another is equal and opposite to the angular impulse exerted back.
- System Selection: A system may be selected so that the total angular momentum of that system is constant (useful in analysis).
- Non-Rigid Systems: The angular speed of a non-rigid system may change without the total angular momentum changing, if mass is moved closer to or farther from the axis (e.g., figure skater pulling in arms).
- Impulse Relation: If the total angular momentum of a system changes, that change is equal to the angular impulse exerted on the system.
Relevant Equations:
Angular Momentum of a rigid body:
$ L = I \omega $
Conservation Condition:
$ I_i \omega_i = I_f \omega_f $
Net torque and angular momentum:
$ \tau_{net} = \dfrac{dL}{dt} $
Angular Impulse–Momentum relation:
$ J_{\theta} = \Delta L $
Example :
A skater spins with arms extended at \( 2 \, \text{rad/s} \). When she pulls in her arms, her moment of inertia reduces to half. Find her new angular velocity.
▶️Answer/Explanation
Conservation: \( I_i \omega_i = I_f \omega_f \).
Substitute: \( I_f = \tfrac{1}{2} I_i \), \( \omega_i = 2 \).
So, \( I_i \cdot 2 = \tfrac{1}{2} I_i \cdot \omega_f \) → \( \omega_f = 4 \, \text{rad/s} \).
Final Answer: Her angular velocity doubles to 4 rad/s.
Example :
Two astronauts (each of mass 80 kg) are initially at rest in space, holding onto each other at a rotating tether. They push apart, applying equal and opposite torques. Describe the angular momentum change of the system.
▶️Answer/Explanation
By Newton’s Third Law in rotational form, the angular impulse one exerts on the other is equal and opposite.
Thus, total angular momentum of the two-astronaut system is conserved (net external torque = 0).
Individually, each astronaut’s angular momentum changes, but the system’s total remains constant.
Key Point: Internal torques cannot change the system’s total angular momentum.
System Selection & Angular Momentum
Main Points:
Conservation Principle:
Angular momentum is conserved in all interactions.
Zero Net External Torque:
If the net external torque on a selected object or rigid system is zero, then the total angular momentum of that system remains constant.
Nonzero Net External Torque:
If the net external torque on a selected system is nonzero, angular momentum is transferred between the system and its environment.
System Selection:
The choice of system determines whether forces/torques are considered internal or external, which in turn decides if angular momentum is conserved.
Example:
A bicycle wheel is spinning freely in space with no external torques. Now imagine the same wheel spinning on Earth with axle friction. Will angular momentum be conserved in both cases?
▶️Answer/Explanation
Case 1 (Wheel in space): Net external torque = 0 → Angular momentum is constant.
Case 2 (Wheel on Earth): Friction at the axle provides an external torque → Angular momentum decreases over time as energy is transferred to the environment (heat).
Key Takeaway: The selection of the system and presence of external torque decide whether angular momentum is conserved.