AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 5: Torque & Rotational DynamicsRotational InertiaParallel Axis Theorem

Parallel Axis Theorem (College Board AP® Physics 1: Algebra-Based)

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Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Parallel axis theorem

  • The rotational inertia of a rigid system can change depending on the orientation of its rotational axis

    • The minimum rotational inertia in any plane is always when the axis passes through the system's center of mass

  • A rigid system may not always rotate about an axis passing through its center of mass

    • As a result, the rotational inertia will always be greater than if the axis passed through the system's center of mass

  • The rotational inertia about an axis that is parallel to any axis passing through the center of mass can be calculated using the parallel axis theorem

I apostrophe space equals space I subscript c m end subscript space plus thin space M d squared

  • Where:

    • I subscript c m end subscript = rotational inertia about an axis passing through the system’s center of mass

    • I apostrophe = rotational inertia about an axis parallel to I subscript c m end subscript

    • M = mass of the system, in kg

    • d = distance between the rotational axis and the system’s center of mass

  • This means that the rotational inertia of a system will be the same when rotating about any parallel axis at an equal distance from the center of mass

Rotational inertia around a parallel axis

Diagram showing an irregular shaped object with a yellow cylindrical rod which is parallel to its center of mass. Variables I', Icm, and d indicate distances and axes.
The rotational inertia about another axis that is parallel to the axis through the center of mass, at a distance d from the object’s center of mass, can be determined using the parallel axis theorem

Worked Example

The rotational inertia of a rod of mass M and length Lrotating about its center of mass is 1 over 12 M L squared. Which of the following describes the rotational inertia of the rod rotating about one of its ends?

A 1 half M L squared

B 1 third M L squared

C 1 over 6 M L squared

D 1 over 12 M L squared

The correct answer is B

Answer:

Step 1: Determine the distance between the center of mass and the parallel axis

d space equals space L over 2

Step 2: Apply the parallel axis theorem to the rod

I apostrophe space equals space I subscript c m end subscript space plus thin space M d squared

I apostrophe space equals space 1 over 12 M L squared space plus space M open parentheses L over 2 close parentheses squared

I apostrophe space equals space 1 over 12 M L squared space plus space 1 fourth M L squared

I apostrophe space equals space 1 third M L squared

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    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.