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AP®PhysicsCollege BoardPhysics 1: Algebra-BasedStudy GuidesUnit 5: Torque & Rotational DynamicsNewton’s First & Second Law in Rotational FormNewton’s Second Law in Rotational Form

Newton’s Second Law in Rotational Form (College Board AP® Physics 1: Algebra-Based)

Study Guide

Katie M

Written by: Katie M

Reviewed by: Caroline Carroll

Newton’s second law in rotational form

The rate of change of angular velocity, or angular acceleration, of a rigid system is directly proportional to the net torque exerted on the rigid system and is in the same direction

  • It is defined by the equation:

alpha subscript s y s end subscript space equals space fraction numerator sum tau over denominator I subscript s y s end subscript end fraction space equals space tau subscript n e t end subscript over I subscript s y s end subscript

  • Where:

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

    • tau subscript n e t end subscript = net torque exerted on the system, measured in straight N times straight m

    • I subscript s y s end subscript = rotational inertia of the system, measured in kg times straight m squared

  • Newton's second law states that angular acceleration is:

    • directly proportional to the net torque exerted on the rigid system

    • inversely proportional to the rotational inertia of the rigid system

  • Therefore, the angular velocity of a rigid system will

    • change if the net torque exerted on that system is not equal to zero

    • remains constant if the net torque exerted on that system is equal to zero

Solving problems involving linear and rotational motion

  • A rigid system may have multiple forces and torques acting on it, for example

    • a beam suspended by a rope

    • pulley-mass systems

  • Torques and linear forces act independently

    • Therefore, to fully describe such a system, calculations involving linear and rotational quantities may need to be performed independently

  • These types of questions usually involve the following concepts and equations

    • Net force is zero for systems in translational equilibrium: sum for i of space stack F subscript i with rightwards arrow on top space equals space 0

    • Net torque is zero for systems in rotational equilibrium: sum tau subscript i space equals space 0

    • Unbalanced forces produce linear acceleration: F subscript n e t end subscript space equals space m a

    • Unbalanced torques produce angular acceleration: tau subscript n e t end subscript space equals space I alpha

    • The relationship between tangential and angular velocity: v subscript T space equals space r omega

    • The relationship between tangential and angular acceleration: a subscript T space equals space r alpha

Comparison of linear and rotational variables in Newton's second law

linear variable

rotational variable

force F

torque tau

mass m

rotational inertia I

acceleration a

angular acceleration alpha

Newton's second law F space proportional to space a

Newton's second law tau space proportional to space alpha

F subscript n e t end subscript space equals space m a

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

Worked Example

A block of mass m is attached to a string that is wrapped around a cylindrical pulley of mass M and radius R, as shown in the diagram.

The rotational inertia of the cylindrical pulley about its axis is 1 half M R squared

Diagram showing a cylindrical pulley of mass M and radius R, with a mass m hanging from its edge.

When the block is released, the pulley begins to turn as the block falls.

Derive an expression for the acceleration of the block in terms of m, M and g.

Answer:

Step 1: Analyze the scenario

  • There is linear motion due to the forces exerted on the block from:

    • the weight of the block

    • the tension in the string

  • There is rotational motion due to the torque exerted on the pulley from the tension in the string at the radius

Diagram showing rotational motion with a cylindrical pulley of radius R and mass M, and linear motion with a mass m hanging from a string. Equations for both motions provided.

Step 2: Apply Newton's second law to the motion of the block

  • A net force produces a linear acceleration

F subscript n e t end subscript space equals space m a

m g space minus space T space equals space m a eq. (1)

Step 3: Apply Newton's second law to the rotation of the pulley

  • A net torque produces angular acceleration

    • For radius R

tau space equals space I alpha

T R space equals space I alpha

Step 4: Write the expression in terms of linear acceleration

  • The relationship between angular acceleration and tangential acceleration of the pulley is:

alpha space equals space a over R

  • Substitute this into the previous equation:

T R space equals space I a over R

Step 5: Substitute the rotational inertia into the expression

  • The rotational inertia of the cylinder is I space equals space 1 half M R squared

T R space equals space open parentheses 1 half M R squared close parentheses a over R

T space equals space open parentheses 1 half M R squared close parentheses a over R squared

T space equals space 1 half M a eq. (2)

Step 6: Write the final expression for linear acceleration

  • Substitute eq. (2) into eq. (1)

m g space minus space open parentheses 1 half M a close parentheses space equals space m a

m g space equals space m a space plus space 1 half M a

m g space equals space a open parentheses m space plus space 1 half M close parentheses

a space equals space fraction numerator m g over denominator m space plus space 1 half M end fraction

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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.