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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 6: Energy & Momentum of Rotating SystemsConservation of Angular MomentumConservation of Angular Momentum

Conservation of Angular Momentum (College Board AP® Physics 1: Algebra-Based)

Study Guide

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Conservation of angular momentum

  • As with linear momentum, angular momentum is always conserved

  • The principle of conservation of angular momentum states:

The total angular momentum of an isolated system remains constant unless acted on by a net external torque

  • Therefore, for an interaction between objects in an isolated system:

The total angular momentum before the interaction is equal to the total angular momentum after the interaction

  • Mathematically, this can be written as:

sum L subscript i space equals space sum L subscript f

  • Where:

    • L subscript i = initial angular momentum (before the interaction), in kg times straight m squared divided by straight s

    • L subscript f = final angular momentum (after the interaction), in kg times straight m squared divided by straight s

Worked Example

A bicycle tire of mass M and radius R is mounted horizontally, allowing it to spin about a frictionless axis. The tire is initially at rest before a dart of mass m strikes it with velocity v at an angle of theta as shown in the diagram.

Diagram showing a top-down view of a wheel with mass M, radius R, and an arrow at angle θ to the radius. A dart of mass m moves tangentially with velocity v.

The rotational inertia of the dart and wheel is open parentheses m space plus thin space M close parentheses R squared.

Derive an expression for the angular velocity of the tire after the dart strikes it in terms of m, M, R and theta.

Answer:

Step 1: Analyze the scenario

  • Before the collision:

    • the tire of mass M is stationary

    • the dart of mass m moves with linear velocity v

  • After the collision:

    • the tire and dart move as one rigid body of rotational inertia open parentheses m space plus space M close parentheses R squared and angular velocity omega

Step 2: Determine the angular momentum of the dart

  • Just before the dart strikes the tire

    • it is located at a distance of R from the axis of rotation

    • its velocity makes an angle of theta with this distance

  • Therefore, using the equation for a particle's angular momentum, the dart can be modeled as a particle with an initial angular momentum of:

L subscript i space equals space m v R space sin space theta

Step 3: Determine the angular momentum of the dart-tire system

  • When the dart strikes the tire, they become one system with an angular momentum of:

L subscript f space equals space I omega space equals space open parentheses m space plus space M close parentheses R squared omega

Step 4: Apply conservation of angular momentum

  • Conservation of angular momentum: the total angular momentum before the collision = the total angular momentum after the collision

sum L subscript i space equals space sum L subscript f

m v R space sin space theta space equals space open parentheses m plus M close parentheses R squared omega

Step 5: Rearrange for the angular velocity omega

omega space equals space fraction numerator m v R space sin space theta over denominator open parentheses m plus M close parentheses R squared end fraction space equals space fraction numerator m v space sin space theta over denominator open parentheses m plus M close parentheses R end fraction

Angular impulse & total angular momentum

The angular impulse exerted by object (or system) A on object (or system) B is equal and opposite to the angular impulse exerted by object (or system) B on object (or system) A

  • If the total angular momentum of a system changes, that change is equivalent to the angular impulse exerted on the system

angular space impulse space equals space increment L

  • Where:

    • angular impulse is measured in straight N times straight m times straight s

    • increment L = change in angular momentum of the system, in kg times straight m squared divided by straight s

  • When two objects, A and B, rotating about the same axis interact:

    • object A exerts a torque on object B of magnitude tau subscript A

    • object B exerts a torque on object A of magnitude tau subscript B

    • the torques are equal in magnitude and opposite in direction tau subscript A space equals space minus tau subscript B

    • the torques, and therefore angular impulses, only act for the duration of the interaction

  • Since the angular impulse is delivered during the interaction, the value of time increment t is equal for each object, hence the change in momentum is:

increment L space equals space tau increment t

  • Therefore, the change in angular momentum of A is equal to the change in angular momentum of B:

increment L subscript A space equals space minus increment L subscript B

Third law torque pair during an interaction

Diagram of a wrench and bolt demonstrating Newton's third law for torque. When a torque τA is exerted by a wrench on a bolt, the bolt exerts an equal torque τB on the wrench in the opposite direction. This is equivalent to the corresponding forces acting at a distance r from the axis of rotation.
According to Newton's third law, the torque exerted by object A (the wrench) on object B (the bolt) is equal to the torque exerted by B on A and opposite in direction. This is equivalent to the corresponding forces acting at a distance r from the axis of rotation

Examiner Tips and Tricks

Make sure you understand that Newton's third law in rotational form only applies when a force is applied at a distance r from a rotational axis. As a result, the Newton's third law force pair acts at the same point (i.e. at a distance r). For example, consider a sphere:

  • when a force is exerted along its rotational axis, only a linear impulse will act about its center of mass

  • when a force is exerted along its circumference (i.e. a distance r from its rotational center), both a linear impulse and an angular impulse will act about its center of mass

Left: When a force is applied along the center of mass of a sphere, it produces linear impulse only as there is no rotation Right: When a force is applied along the circumference of a sphere, it produces both linear and angular impulse as there is rotation

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    Author
    Reviewer
    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

    Caroline Carroll

    Author: Caroline Carroll

    Expertise: Physics Subject Lead

    Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.