Study GuidesExam QuestionsPast Papers
AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 5: Torque & Rotational DynamicsRotational KinematicsConnecting Linear & Rotational Motion

Connecting Linear & Rotational Motion (College Board AP® Physics 1: Algebra-Based)

Study Guide

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Connecting linear & rotational motion

  • The rotational quantities that describe a point on a rotating rigid body can be related to their corresponding linear quantities

Summary of linear and angular variables

variable

linear variable

angular variable

displacement

s

theta

velocity

v

omega

acceleration

a

alpha

Linear and angular displacement

  • For a point on a rigid system that rotates about a fixed axis of rotation, the linear distance traveled by the point is given by:

increment s space equals space r increment theta

  • Where:

    • increment s = linear distance traveled by the point, in straight m

    • increment theta = angular displacement of the point, in rad

    • r = distance from a fixed axis of rotation, in straight m

Diagram showing rotational motion with labeled elements: axis of rotation, radius (r), angular displacement (Δθ), linear distance (arc length Δs), and direction of rotation.
Using the definition of a radian, the linear distance s traveled by a point on a rotating rigid system can be related to the angular displacement Δθ

Linear and angular velocity

  • The linear velocity of a point on a rotating rigid body is equal to the linear distance traveled by the point divided by the time increment t

v space equals space fraction numerator increment s over denominator increment t end fraction space equals space fraction numerator r increment theta over denominator increment t end fraction

  • The angular velocity of the system is equal to the rate of change of angular displacement

omega space equals space fraction numerator increment theta over denominator increment t end fraction

  • Therefore, the linear speed of a point is related to the angular speed of the system by the equation:

v space equals space r omega

  • Where:

    • v = linear speed of a point, in straight m divided by straight s

    • omega = angular speed of the system, in rad divided by straight s

    • r = distance from a fixed axis of rotation, in straight m

  • For a rigid system, all points within that system have the same angular velocity

    • However, not all points have the same linear velocity

  • The relationship above tells us the linear velocity of a point:

    • increases with distance from the axis of rotation

    • has a maximum value at the radius of the system

    • is zero at the axis of rotation

  • The linear velocity of a point on a rigid rotating body is also known as its tangential velocity

    • This is because the instantaneous direction of the velocity is always at a tangent to the direction of rotation

Diagram of a rotating circle showing angular velocity (ω) as constant. Linear velocity (v) varies with distance from the axis, greatest at the circumference.
All points on the rigid object will have the same angular velocity, but not the same linear velocity

Linear and angular acceleration

  • The tangential acceleration of a point on a rotating rigid body is equal to the rate of change of linear velocity at that point

a subscript T space equals space fraction numerator increment v over denominator increment t end fraction space equals space fraction numerator r increment omega over denominator increment t end fraction

  • The angular acceleration of the system is equal to the rate of change of angular velocity

alpha space equals space fraction numerator increment omega over denominator increment t end fraction

  • Therefore, the tangential acceleration of a point is related to the angular acceleration of the system by the equation:

a subscript T space equals space r alpha

  • Where:

    • a subscript T = tangential acceleration of a point, in straight m divided by straight s squared

    • alpha = angular acceleration of the system, in rad divided by straight s squared

    • r = distance from a fixed axis of rotation, in straight m

  • For a rigid system, all points within that system have the same angular acceleration

Examiner Tips and Tricks

While there are many similarities between the angular quantities used in this topic and the angular quantities used in the circular motion topic, make sure you are clear on the distinctions between the two, for example, angular acceleration and centripetal acceleration are not the same thing!

Last updated:

You've read 0 of your 5 free study guides this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself

    Did this page help you?

    Previous:Angular AccelerationNext:Rotational Kinematics Equations
    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.