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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 5: Torque & Rotational DynamicsTorqueDefining Torque

Defining Torque (College Board AP® Physics 1: Algebra-Based)

Study Guide

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

What is torque?

  • Torque is the rotational analog of force

  • In the same way that a net force causes linear acceleration, a net torque causes angular acceleration

  • A torque is produced when a force is applied at a distance from an axis of rotation and causes a change in rotational motion

    • The component of the force that causes the torque is perpendicular to the lever

    • The perpendicular distance from the axis of rotation to the line of action of the force is called the lever arm

  • This means that a force applied at the pivot, or axis of rotation, will not produce a torque

Torque exerted on a closing door

Diagram depicting a door being closed with labeled parts: hinge (axis of rotation), door, lever arm, applied force (red arrow), torque (curved arrow), and rotational motion.
When a force is applied to a lever (e.g. a door), a torque is exerted about the axis of rotation (i.e. the door's hinge)

  • The magnitude of a torque depends on

    • the magnitude of the applied force

    • the distance from the axis of rotation to the point of application of the force

    • the angle between the position vector and the force vector

Diagram showing three scenarios of force applied to a door with varying distances and angles from the axis of rotation to illustrate torque.
The largest magnitude of torque is produced when a force is applied furthest from the axis of rotation and perpendicular to the lever

Torque equation

  • Torque can be defined as:

The product of the force and the perpendicular distance from the axis of rotation (lever arm)

  • The magnitude of the torque exerted on a rigid system by a force is:

tau space equals space r F subscript perpendicular space equals space r F space sin space theta

  • Where:

    • tau = magnitude of the torque, in straight N times straight m

    • F subscript perpendicular = the component of the force that is perpendicular to the lever arm

    • F = magnitude of the applied force, in straight N

    • r = lever arm, in straight m

    • theta = angle between the applied force and the lever arm in degree

Torque of a perpendicular force

Diagram of a spanner illustrating torque, showing pivot point, applied force direction, distance from pivot to force, and perpendicular force application.
A maximum torque is produced when a force applied is at right angles to the distance between the pivot and the point of application of the force.

Torque of a non-perpendicular force

Diagram of a bicycle pedal rotating around an axis, with labeled parts: axis of rotation, line of action, force (F), radius (r), and angle (θ) between force and arm of the mechanism.
The torque applied by a cyclist on a bicycle pedal is equal to the product of the applied force F and the perpendicular distance between the line of action of the force and the axis of rotation r sin θ

  • The maximum torque that can be exerted on a rigid system is when the force is perpendicular open parentheses theta space equals space 90 degree close parentheses to the lever arm

tau space equals space F r

  • As a result, the same force is less effective when applied at non-perpendicular angles

Effect of angle on torque

Diagram showing the relationship between force, angle, and torque on a wrench. As the angle decreases from 90 degrees, torque reduces, eventually approaching zero.
The force produces a maximum torque when applied perpendicular to the arm of the wrench. The same force is less effective when applied at non-perpendicular angles

Worked Example

The forces acting on a bicycle pedal at different positions during a ride are shown in the diagram below. The distance from the pedal to the axis of rotation is 24 cm.

In which of the following positions is the magnitude of the torque the greatest?

Diagrams A, B, C, and D show gears with varying forces (400 N, 90 N, 75 N, 150 N) applied at different angles (60°, 30°) to the attached levers.

The correct answer is B

Answer:

Step 1: Analyze the scenario

  • The magnitude of torque on a rigid system is:

tau space equals space F r space sin space theta

  • The maximum torque is exerted when theta space equals space 90 degree, where theta is the angle between the line of action of the force and the lever arm

  • In each case, the lever arm is the same, r = 24 cm = 0.24 m

Step 2: Eliminate incorrect options

  • In position A: F = 400 N and theta = 180°, so:

tau space space equals space 400 cross times sin space 180 degree space equals space 400 cross times 0 space equals space 0

  • This means when the component of force is parallel to the lever arm, no torque is exerted on the pedal

  • Therefore, when the pedal is in this position, no amount of pushing down will produce any change in its rotational motion

Step 3: Deduce the correct answer

  • In position B: F = 90 N and theta = 90°, so:

tau space equals space 0.24 cross times 90 cross times sin space 90 space equals space 21.6 space straight N times straight m

  • In position C: F = 75 N and theta = 60°, so:

tau space equals space 0.24 cross times 75 cross times sin space 60 space equals space 15.6 space straight N times straight m

  • In position D: F = 150 N and theta = 30°, so:

tau space equals space 0.24 cross times 150 cross times sin space 30 space equals space 18.0 space straight N times straight m

  • The largest torque occurs in position B

Examiner Tips and Tricks

You should know that torque is a vector quantity, however, in the AP Physics 1 exam, you will only need to consider whether it produces clockwise or anti-clockwise motion, and then use vector conventions to calculate the magnitude of torque accordingly. The direction of torque is beyond the scope of the course and will not be tested.

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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.