Rotational Motion

Introduction

Rotational motion is a fundamental concept in physics, describing the movement of objects around a fixed axis. This topic introduces angular quantities, their relationships to linear motion, and the energy associated with rotating bodies.

The Radian

Definition and Conversion

• Radian is the standard unit for measuring angles in physics, defined as the ratio of arc length to radius.

• Conversion from degrees to radians:

• Common conversions:

  • rad

  • rad

  • rad

  • rad

Angular Displacement

Definition

• Angular displacement () is the angle through which an object rotates, measured in radians.

• 1 radian = 57.3°; 2π radians = 360°.

Angular Velocity

Definition and Units

• Angular velocity () is the rate of change of angular displacement.

• Average angular velocity:

• Instantaneous angular velocity:

• Units: radians per second (rad/s).

Angular Acceleration and Sign Convention

Definition and Direction

• Angular acceleration () is the rate of change of angular velocity.

• Average angular acceleration:

• Units: radians per second squared (rad/s²).

• Sign convention:

  • Counterclockwise rotation: positive ()

  • Clockwise rotation: negative ()

Linear and Angular Equations

Comparison of Linear and Rotational Kinematics

Relationship Between Linear and Angular Quantities

Key Equations

• Linear speed:

• Radial (centripetal) acceleration:

• Tangential acceleration:

Rotational Energy

Kinetic Energy of Rotation

• A rotating rigid body has kinetic energy due to its motion.

• Total rotational kinetic energy:

• Moment of inertia () quantifies the distribution of mass relative to the axis of rotation:

• For a system of particles:

Moments of Inertia for Common Bodies

Rotation About a Moving Axis

Total Kinetic Energy

• When both the center of mass and the axis of rotation are moving, total kinetic energy is:

• This includes translational kinetic energy of the center of mass and rotational kinetic energy about the center of mass.

Examples and Applications

Example 1: Washing Machine Spin Cycle

• Given two angular speeds, calculate ratios of radial force and tangential speed, and express maximum values in terms of .

• Key equations:

  • Radial force:

  • Tangential speed:

  • Radial acceleration:

Example 2: Rolling Marble in a Bowl

• Analyze energy conservation for a marble rolling with and without slipping.

• Compare maximum heights and speeds for frictionless and rough surfaces.

Example 3: Yo-Yo Unwinding

• Find the speed of the center of mass after dropping a distance using energy conservation and rotational dynamics.

Important Concepts

• Angular velocity and acceleration calculations, including degree-to-radian conversions.

• Relationships between angular and linear equations.

• Rotational energy and moment of inertia.

• Rotation about a moving axis and total kinetic energy.

Additional info: These notes are based on Chapter 9 of a College Physics course, focusing on rotational motion, and include both conceptual explanations and worked examples relevant for exam preparation.