Rotation of Rigid Bodies

Introduction

Many everyday objects, such as wind turbines, CDs, ceiling fans, and Ferris wheels, involve the rotation of rigid bodies. In physics, a rigid body is an idealization where the distances between all pairs of points in the object remain constant during motion. This chapter explores the kinematics and dynamics of rotational motion, drawing analogies to linear motion.

• Rigid body: An object that does not deform during motion.

• Rotational motion: Motion of a body about a fixed axis.

Rotational Kinematics

Angular Position, Displacement, and Units

Rotational motion is described using angular variables analogous to linear variables.

• Angular position (θ): The angle (in radians) that specifies the orientation of a reference line fixed in the rotating body relative to a fixed axis.

• Arc length (s): The distance along the circle, related to θ by .

• Units: 1 revolution = radians = 360°.

Equation:

Angular Velocity (ω)

Angular velocity describes how fast the angular position changes with time.

• Average angular velocity:

• Instantaneous angular velocity:

• SI units: radians per second (rad/s)

• Direction: Positive for counterclockwise, negative for clockwise rotation.

Angular Acceleration (α)

Angular acceleration measures how quickly the angular velocity changes.

• Average angular acceleration:

• Instantaneous angular acceleration:

• SI units: radians per second squared (rad/s2)

Vector Nature of Angular Quantities

• Angular velocity and angular acceleration are vectors along the axis of rotation (right-hand rule).

• If and point in the same direction, the object speeds up; if opposite, it slows down.

Period and Frequency

• Period (T): Time for one complete revolution.

• Frequency (ν): Number of revolutions per second.

Equations:

Rotational Kinematic Equations (Constant Angular Acceleration)

These equations are analogous to linear kinematics:

Connections Between Linear and Rotational Quantities

• Linear velocity (tangential):

• Tangential acceleration:

• Centripetal (radial) acceleration:

• Total acceleration:

• Angle with tangential direction:

Rotation of Extended Objects

Describing Extended Objects

• Each mass point in a rigid body moves in a circle about the axis of rotation.

• All points share the same angular velocity () and angular acceleration (), but have different linear velocities depending on their distance from the axis.

Kinetic Energy of Rotation

• Kinetic energy of a mass point:

• Total rotational kinetic energy:

• Moment of inertia (I): (sum over all mass points)

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion, analogous to mass in linear motion.

• For a point mass:

• For an extended object:

• Depends on: Mass distribution and axis of rotation.

Examples of Moments of Inertia

Parallel Axis Theorem

The parallel axis theorem allows calculation of the moment of inertia about any axis parallel to one through the center of mass:

• = moment of inertia about the parallel axis

• = moment of inertia about the center of mass axis

• = total mass

• = distance between axes

Gravitational Potential Energy of an Extended Body

• The gravitational potential energy is as if all mass were concentrated at the center of mass:

Uses of Energy Conservation

• Mechanical energy conservation applies to rotating bodies:

• For a rotating object descending under gravity, potential energy converts to rotational kinetic energy.

Summary Table: Linear vs. Rotational Quantities

Example Application

Example: Calculate the moment of inertia of a solid cylinder (mass , radius ) about its central axis.

• Use

• If kg, m, then kg·m2

Additional info: The notes also emphasize the importance of angular momentum conservation and the analogy between linear and rotational dynamics, which are foundational for understanding more advanced topics in physics such as torque, equilibrium, and rotational dynamics.