Rotation of Rigid Bodies

Introduction

• Many everyday objects, such as airplane propellers, revolving doors, ceiling fans, and Ferris wheels, involve the rotation of rigid bodies.

• In physics, a rigid body is an idealization where the object does not deform during rotation; all points maintain fixed distances from each other.

• Real-world objects may stretch or twist, but for introductory analysis, we assume perfect rigidity.

Angular Coordinate

Definition and Measurement

• The angular coordinate (θ) specifies the rotational position of a point or object about a fixed axis.

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

Units of Angles

• The radian is the SI unit for measuring angles.

• One radian is the angle subtended at the center of a circle by an arc whose length equals the radius: .

• One complete revolution: radians.

• Angle in radians: , where s is the arc length and r is the radius.

Angular Velocity

Average and Instantaneous Angular Velocity

• The average angular velocity about the z-axis is .

• The instantaneous angular velocity is the limit as :

• Angular velocity can be positive or negative, depending on the direction of rotation (counterclockwise is positive by convention).

• The subscript z indicates rotation about the z-axis.

Angular Velocity as a Vector

• Angular velocity is a vector, with direction given by the right-hand rule: curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of .

• For rotation along the z-axis:

  • : positive z-direction

  • : negative z-direction

Example: Rotational Motion in Bacteria

• Escherichia coli bacteria swim by rotating their flagella like propellers.

• Flagella rotate at angular speeds of 200–1000 rev/min (about 20–100 rad/s), with variable angular acceleration.

Angular Acceleration

Definition

• The average angular acceleration is the change in angular velocity divided by the time interval:

• The instantaneous angular acceleration is:

Angular Acceleration as a Vector

• If and point in the same direction, the rotation is speeding up.

• If and point in opposite directions, the rotation is slowing down.

Rotation with Constant Angular Acceleration

Rotational Kinematic Equations

• For constant angular acceleration, the equations mirror those for straight-line motion:

Relating Linear and Angular Kinematics

Linear Speed and Acceleration

• A point at a distance r from the axis of rotation has a linear speed:

• Tangential acceleration:

• Centripetal (radial) acceleration:

The Importance of Using Radians

• Always use radians (not degrees) when relating linear and angular quantities in equations.

• For example, is valid only if is in radians.

Rotational Kinetic Energy

Definition and Formula

• The rotational kinetic energy of a rigid body is:

• Moment of inertia () is the rotational analog of mass, quantifying how mass is distributed relative to the axis of rotation.

• Calculated as: (for discrete masses) or (for continuous mass distributions).

• SI unit:

Moment of Inertia

Dependence on Mass Distribution

• Moment of inertia depends on both the mass and how far the mass is from the axis of rotation.

• Moving mass closer to the axis decreases (easier to rotate); moving mass farther increases (harder to rotate).

Example: Bird's Wing

• Hummingbirds have small wings (small ), allowing rapid flapping (up to 70 beats/s).

• Andean condors have large wings (large ), resulting in slower flapping (about 1 beat/s).

Moments of Inertia of Common Objects

• Standard formulas for various shapes (from Table 9.2):

Gravitational Potential Energy of an Extended Object

• The gravitational potential energy of an extended object is as if all its mass were concentrated at its center of mass:

• Applications include athletic techniques (e.g., high jump) that minimize the required energy by manipulating the center of mass.

The Parallel-Axis Theorem

• The parallel-axis theorem relates the moment of inertia about any axis parallel to one through the center of mass:

• Where is the moment of inertia about the parallel axis, is about the center of mass, is the total mass, and is the distance between axes.

Moment of Inertia Calculations

• For continuous mass distributions, the moment of inertia is calculated by integration:

• Here, is the mass density, and the integral is over the object's volume.

• Applications include determining the Earth's internal mass distribution by analyzing satellite orbits.