Chapter 10: Dynamics of Rotational Motion

Learning Goals

This chapter explores the physical principles governing rotational motion, focusing on torque, angular momentum, and the dynamics of rotating bodies. Students will learn to:

• Define and calculate torque produced by a force.

• Analyze how net torque affects rotational motion.

• Examine bodies that both rotate and translate through space.

• Solve problems involving work and power for rotating bodies.

• Understand conservation of angular momentum, even when a body's shape changes.

Introduction to Rotational Motion

Rotational motion is observed in everyday phenomena, such as juggling pins or spinning wheels. Understanding what initiates or halts rotation requires new concepts like torque and angular momentum.

• Torque is the rotational equivalent of force.

• Angular momentum describes the rotational analog of linear momentum.

Torque

Torque is a measure of the tendency of a force to rotate an object about an axis. The effectiveness of a force in producing rotation depends on its magnitude, direction, and the distance from the axis of rotation.

• Line of action: The straight line along which the force acts.

• Lever arm: The perpendicular distance from the axis of rotation to the line of action of the force.

• Torque formula: The torque () produced by a force () at a distance () from the axis is:

• Where is the angle between the force vector and the position vector from the axis.

• Alternatively, , where is the lever arm.

Example: When loosening a bolt with a wrench, applying force farther from the axis increases the torque and is more effective.

Calculating Torque

There are several equivalent ways to calculate torque:

Where is the tangential component of the force.

Torque as a Vector

Torque is a vector quantity, defined by the vector product (cross product) of the position vector and the force vector:

• The direction of is given by the right-hand rule: curl the fingers of your right hand from toward ; your thumb points in the direction of .

Rotational Analog of Newton's Second Law

For a rigid body rotating about a fixed axis, the net torque is related to angular acceleration:

• is the moment of inertia of the body about the axis.

• is the angular acceleration.

Example: Tightening or loosening a screw requires applying a torque to produce angular acceleration.

Internal vs. External Torques

Only external torques affect the rotation of a rigid body. Internal forces (action-reaction pairs) produce equal and opposite torques that cancel each other.

Problem-Solving Strategy for Rotational Dynamics

To solve rotational dynamics problems:

1. Identify the relevant concept ().

2. Sketch the situation and indicate the axis of rotation.

3. Draw free-body diagrams, labeling all forces and dimensions.

4. Choose coordinate axes and indicate the positive sense of rotation.

5. Determine if the body undergoes translational, rotational, or both types of motion.

6. Apply Newton's second law for translation () and/or rotation ().

7. Express any geometric relationships between motions.

8. Solve the equations for the desired variables.

Rotation About a Moving Axis

The kinetic energy of a rigid body that both translates and rotates is:

• is the total mass, is the velocity of the center of mass, is the moment of inertia about the center of mass, and is the angular velocity.

Example: The motion of a tossed baton can be represented as translation of the center of mass plus rotation about the center of mass.

Rolling Without Slipping

For a wheel rolling without slipping:

• The velocity of the center of mass is related to angular velocity by .

• The motion is a combination of translation and rotation.

Example: A rolling wheel's point of contact with the ground is instantaneously at rest.

Rolling With Slipping

If the wheel slips, . Frictional forces and energy loss become significant.

Combined Translation and Rotation: Dynamics

The acceleration of the center of mass is determined by:

Rotational motion about the center of mass is described by:

• Valid if the axis through the center of mass is a symmetry axis and does not change direction.

Work and Power in Rotational Motion

Work done by a torque is:

The total work done equals the change in rotational kinetic energy:

Power due to a torque is:

• is the angular velocity.

Example: In a helicopter rotor, the engine does positive work while air resistance does negative work; if net work is zero, kinetic energy remains constant.

Angular Momentum

Angular momentum () for a particle is:

• For a rigid body rotating about a symmetry axis:

• is the moment of inertia, is angular velocity.

For a system of particles:

Conservation of Angular Momentum

If the net external torque on a system is zero, angular momentum is conserved:

• Example: A spinning cat twists its body to land feet first, conserving total angular momentum.

Gyroscopes and Precession

When a gyroscope spins, its axis of rotation can change direction—a motion called precession. The change in angular momentum is perpendicular to the applied torque, causing the axis to trace a circular path.

• Non-rotating gyroscope: falls due to torque from gravity.

• Rotating gyroscope: precesses around the pivot, maintaining its axis due to conservation of angular momentum.

Summary Table: Key Rotational Quantities

Additional info: These notes expand on the provided slides and images, adding definitions, formulas, and examples for clarity and completeness.