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Conservation of angular momentum review

Review how angular momentum is conserved if there is no external torque applied to a system.  Specific scenarios are highlighted: system changing it rotational inertia, orbiting bodies, and rotational collisions.

Key terms

Term (symbol)Meaning
Conservation of angular momentumAngular momentum is constant for a system with no net external torque.


EquationSymbol breakdownMeaning in words
Li=LfLi and Lf are the total initial and final angular momentum.Total initial angular momentum equals the total final angular momentum for a system with no net external torque. Commonly called the conservation of angular momentum.

Applying the conservation of angular momentum

Objects can change their shape and still conserve angular momentum

Angular momentum depends on the rotational velocity of an object, but also its rotational inertia. When an object changes its shape (rotational inertia), its angular velocity will also change if there is no external torque.
An example is when an ice skater spins and changes her rotation velocity by holding her arms outwards or pulling them inwards (see Figure 1 below).
Figure 1. Ice skater bringing her arms inward to decrease rotational inertia. Original image courtesy of Wikimedia Commons, CC BY-SA 3.0.
When she pulls her arms in, her rotational inertia is reduced. Since there is no external net torque on the ice skater, her angular momentum remains constant because her angular velocity magnitude increases.

Orbital systems such as our solar system have angular momentum conserved

A system of planets orbiting a star has no net external torque, so its angular momentum is constant. As a planet travels along an elliptical orbit, its speed reduces when it is further away from the star, and increases its speed as it approaches the star as seen in Figure 2.
Figure 2. Angular momentum is conserved for orbital systems.
This keeps the angular momentum about the star constant by adjusting the variables of v and r for the angular momentum equation below:
This scenario of orbiting objects is discussed with more detail in the next lesson.

Rotational collisions conserve angular momentum

When objects collide without a net external torque, the angular momentum is constant. The two objects exert equal, but opposite angular impulses upon each other to maintain the total angular momentum of the colliding system. An example of this would be a ball colliding with a stick that rotates about its end as shown in Figure 3.
Figure 3. Angular momentum is conserved during collisions.

Learn more

For deeper explanations of angular momentum conservation concepts, see our video about the conservation of angular momentum.
To check your understanding and work toward mastering these concepts, check out our exercises:

Want to join the conversation?

  • female robot amelia style avatar for user 20lawincentsen
    How would you show the graph of angular momentum vs. time of a collision of a ball and stick that rotates at around its end?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Chandler Juego
      I think it depends on what system you are graphing. If you are graphing just the rod alone, then you would show that its angular momentum is initially zero until the moment of collision where its angular momentum equals the angular momentum of the ball prior to the collision; this is only if the ball stopped and transferred all its angular momentum into the rod. However, if your showing a graph of the angular momentum of the ball-rod system, then by conservation of angular momentum, the total angular momentum would be the same prior and after the collision.
      (2 votes)