Rotation of Rigid Bodies – University Physics Chapter 9 Study Notes

Rotation of Rigid Bodies

Introduction

Rotational motion is a fundamental concept in physics, describing how objects spin about an axis. Examples include airplane propellers, revolving doors, ceiling fans, and Ferris wheels. In this chapter, we focus on the ideal case of rigid bodies, which do not deform during rotation.

• Rigid body: An object whose shape does not change during motion.

• Rotation axis: The fixed line about which the body rotates.

• Real-world applications: Engineering (motors, turbines), biology (flagella), sports (discus throw).

Angular Coordinate

The angular coordinate specifies the rotational position of a point or object about a fixed axis. It is measured as the angle from a reference direction (often the +x-axis).

• Angle θ: The measure of rotation from a reference axis.

• Axis of rotation: The line about which the rotation occurs, often passing through the origin and perpendicular to the plane of motion.

• Example: A car's speedometer needle rotates about a fixed axis, and its position is described by θ.

Units of Angles

Angles in rotational motion are measured in radians, which relate arc length to radius.

• Radian: The angle subtended at the center of a circle by an arc equal in length to the radius.

• Conversion: One complete revolution is $360^ ext{o} = 2 ext{π}$ radians.

• Formula: $θ = rac{s}{r}$, where $s$ is arc length and $r$ is radius.

• Example: If $s = r$, then $θ = 1$ radian.

Summary Table: Angle Units

Unit	Definition	Conversion
Degree	1/360 of a full circle	$360^ ext{o} = 2 ext{π}$ radians
Radian	Arc length equals radius	$1$rad$= 57.3^ ext{o}$
Revolution	One full turn	$1$rev$= 2 ext{π}$ radians

Key Equations

• Angle in radians: $θ = rac{s}{r}$

• Arc length: $s = rθ$

Additional info:

• Radians are the standard unit in physics for angular measurements, especially when relating angular and linear quantities.

• Always use radians in equations involving angular velocity, acceleration, or displacement.