Chapter 8: Rotational Motion

Introduction

This chapter explores the fundamental concepts of rotational motion, drawing analogies to linear motion and introducing new quantities such as angular displacement, velocity, acceleration, torque, and rotational inertia. Applications include rolling motion, conservation of angular momentum, and the vector nature of angular quantities.

Angular Quantities

Definition and Measurement

• Rotational motion occurs when all points on an object move in circles around a fixed axis.

• Angle (θ) in radians: , where is the arc length and is the radius.

• 1 revolution = radians.

Angular Displacement, Velocity, and Acceleration

• Angular displacement:

• Average angular velocity:

• Instantaneous angular velocity:

• Average angular acceleration:

• Instantaneous angular acceleration:

Relationship to Linear Quantities

• Each point on a rotating body has both angular velocity () and linear velocity ():

• Objects farther from the axis of rotation move faster (greater for larger ).

• Tangential acceleration:

• Centripetal acceleration:

Correspondence Table

Frequency and Period

• Frequency (f): Number of revolutions per second,

• Period (T): Time for one revolution,

• Frequency is measured in hertz (Hz), where .

Constant Angular Acceleration

The equations of motion for constant angular acceleration mirror those for linear motion, with angular quantities substituted for linear ones.

Rolling Motion (Without Slipping)

• When a wheel rolls without slipping, the point of contact with the ground is instantaneously at rest.

• Relationship between linear and angular speed:

• From different reference frames, the velocity of points on the wheel changes accordingly.

Torque

Definition and Calculation

• Torque (τ): The rotational equivalent of force; it causes objects to rotate.

• Defined as , where is the perpendicular distance from the axis of rotation to the line of action of the force.

• A longer lever arm increases the torque for the same applied force.

Examples

• Using a wrench or a tire iron: a longer handle makes it easier to turn bolts due to increased torque.

• Opening a door: pushing farther from the hinge increases the torque.

Rotational Dynamics; Torque and Rotational Inertia

• Newton's second law for rotation:

• Rotational inertia (I): ; depends on mass distribution and axis location.

• Objects with more mass farther from the axis have greater rotational inertia.

Rotational Inertia Table (Sample)

Solving Problems in Rotational Dynamics

1. Draw a diagram.

2. Define the system.

3. Draw free-body diagrams for each object, showing all forces and their points of application.

4. Identify the axis of rotation and calculate torques about it.

5. Apply Newton's second law for rotation ().

6. Apply Newton's second law for translation and other relevant principles.

7. Solve the equations.

8. Check units and order of magnitude.

Rotational Kinetic Energy

• Kinetic energy of rotation:

• Expressed in rotational terms:

• For objects with both translational and rotational motion:

• When using energy conservation, include both translational and rotational kinetic energy.

Angular Momentum and Its Conservation

• Angular momentum (L):

• Total torque equals the rate of change of angular momentum:

• If net torque is zero, angular momentum is conserved:

• Systems can change their rotational inertia (e.g., a figure skater pulling in arms), altering their rotation rate to conserve angular momentum.

Vector Nature of Angular Quantities

• Angular velocity, acceleration, and momentum are vector quantities pointing along the axis of rotation.

• Direction is determined by the right-hand rule: curl fingers in the direction of rotation, thumb points along the vector.

Summary of Key Points

• Angles are measured in radians; radians in a full circle.

• Angular velocity and acceleration describe rotational motion and relate to linear velocity and acceleration.

• Frequency and period describe the timing of rotational cycles.

• Torque is the product of force and lever arm; it causes rotational acceleration.

• Rotational inertia depends on mass distribution and axis location.

• Rotational kinetic energy and angular momentum are essential for analyzing rotating systems.

• Angular momentum is conserved in the absence of external torque.