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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 6: Energy & Momentum of Rotating SystemsAngular Momentum & Angular ImpulseImpulse–Momentum Theorem in Rotational Form

Impulse–Momentum Theorem in Rotational Form (College Board AP® Physics 1: Algebra-Based)

Study Guide

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Rate of change of angular momentum

The rate of change of angular momentum is equal to the net external torque exerted on an object (or system)

  • This can be written as:

tau space equals space fraction numerator increment L over denominator increment t end fraction

  • Where:

    • tau = torque exerted on the system, measured in straight N times straight m

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

    • increment t = time interval, measured in straight s

  • The magnitude of the change in angular momentum is the difference between an object or system's final and initial angular momenta:

change in angular momentum = final angular momentum − initial angular momentum

increment L space equals space L space minus space L subscript 0

  • The equation above can be used in situations where the rotational inertia of the body is not constant

  • When the rotational inertia is constant:

tau subscript n e t end subscript space equals space fraction numerator increment L over denominator increment t end fraction space equals space fraction numerator L space minus space L subscript 0 over denominator increment t end fraction space equals space fraction numerator I omega space minus space I omega subscript 0 over denominator increment t end fraction space equals space fraction numerator I open parentheses omega space minus space omega subscript 0 close parentheses over denominator increment t end fraction space equals space fraction numerator I increment omega over denominator increment t end fraction

  • Since angular acceleration alpha is equal to the rate of change of angular velocity, the equation becomes:

tau subscript n e t end subscript space equals space I alpha

  • Where:

    • I = rotational inertia of the body, measured in kg times straight m squared

    • increment omega = change in angular velocity, measured in rad divided by straight s

    • omega = final angular velocity, measured in rad divided by straight s

    • omega subscript 0 = initial angular velocity, measured in rad divided by straight s

    • alpha = angular acceleration, measured in rad divided by straight s squared

  • More about this equation can also be found in the study guide for Newton’s second law in rotational form

Impulse–momentum theorem in rotational form

The angular momentum of an object (or system) remains constant unless an external net torque acts upon it

  • An angular impulse is exerted when an external torque is applied for a time

  • Therefore, it must also change the angular momentum of the system

  • The rotational form of the impulse-momentum theorem states that the angular impulse exerted on an object or rigid system is equal to the change in angular momentum

angular space impulse space equals space increment L space equals space tau increment t

  • Where:

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

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

    • tau = torque exerted on the system, measured in straight N times straight m

    • increment t = time interval, measured in straight s

  • The equation will appear in this form on your equation sheet

  • In calculations, a more useful form of this equation is

increment L space equals space tau increment t space equals space I open parentheses omega space minus space omega subscript 0 close parentheses

  • Where:

    • I = rotational inertia of the body, measured in kg times straight m squared

    • omega = final angular velocity, measured in rad divided by straight s

    • omega subscript 0 = initial angular velocity, measured in rad divided by straight s

  • This form can be derived from the equation for angular momentum open parentheses L space equals space I omega close parentheses

  • The rotational form of Newton’s second law is a direct result of the rotational form of the impulse-momentum theorem applied to systems with constant rotational inertia

tau subscript n e t end subscript space equals space fraction numerator increment L over denominator increment t end fraction space equals space fraction numerator I increment omega over denominator increment t end fraction space equals space I alpha

  • Where:

    • increment omega = change in angular velocity, measured in rad divided by straight s

    • alpha = angular acceleration, measured in rad divided by straight s squared

  • The rotational form of the impulse-momentum theorem tells us

    • for a given change in angular momentum, a small torque acting over a long time has the same effect as a large torque acting over a short time

    • for a constant torque, applying the torque over a longer time will lead to a greater change in angular momentum

    • for a specified time, a greater torque will lead to a greater change in angular momentum

Examiner Tips and Tricks

The following equation does not appear on the equation sheet:

tau subscript n e t end subscript space equals space fraction numerator increment L over denominator increment t end fraction space equals space fraction numerator I increment omega over denominator increment t end fraction space equals space I alpha

Each part of this equation can be easily derived, however, using the following equations from the equation sheet:

straight capital delta L space equals space tau straight capital delta t

L space equals space I omega

alpha subscript s y s end subscript space equals space tau subscript n e t end subscript over I subscript s y s end subscript

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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.