Rotational Motion, Dynamics, and Equilibrium: Study Guide (Chapters 9–11)

Rotational Quantities and Vector Directions

Introduction

Rotational motion involves quantities analogous to linear motion, such as angular displacement, angular velocity, and angular acceleration. Understanding vector directions is essential for correctly applying the right-hand rule and solving problems involving rotation.

• Angular Displacement (\( \theta \)): The angle through which an object rotates, measured in radians.

• Angular Velocity (\( \omega \)): The rate of change of angular displacement; direction given by the right-hand rule.

• Angular Acceleration (\( \alpha \)): The rate of change of angular velocity.

• Right-Hand Rule: Used to determine the direction of angular quantities; curl fingers in the direction of rotation, thumb points along the vector.

Example: A wheel rotating counterclockwise has an angular velocity vector pointing out of the plane (toward the observer).

Moment of Inertia & Parallel Axis Theorem

Moment of Inertia (\( I \))

The moment of inertia quantifies an object's resistance to changes in rotational motion. It depends on the mass distribution relative to the axis of rotation.

• Discrete Masses: \( I = \sum m_i r_i^2 \)

• Continuous Mass Distribution: \( I = \int r^2 dm \)

• Common Shapes: Use tables for standard objects (e.g., solid disk, hoop, rod).

Parallel Axis Theorem

Used to find the moment of inertia about any axis parallel to one through the center of mass.

• \( I = I_{\text{cm}} + M D^2 \)

• Where \( I_{\text{cm}} \) is the moment of inertia about the center of mass, \( M \) is total mass, and \( D \) is the distance between axes.

Example: Calculating the moment of inertia of a rod about one end using the parallel axis theorem.

Rotational Kinematics (Constant Angular Acceleration)

Equations of Rotational Kinematics

Analogous to linear kinematics, these equations describe rotational motion with constant angular acceleration.

• \( \omega = \omega_0 + \alpha t \)

• \( \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \)

• \( \omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0) \)

Example: A disk starts from rest and accelerates at \( 2\ \text{rad/s}^2 \) for 3 seconds. Find its angular displacement.

Rotational Kinetic Energy, Work, and Work-Energy Theorem

Rotational Kinetic Energy

• \( K_{\text{rot}} = \frac{1}{2} I \omega^2 \)

Rotational Work

• \( W = \tau \theta \) (for constant torque)

Rotational Work-Energy Theorem

• \( W_{\text{net}} = \Delta K_{\text{rot}} \)

Example: Calculating the work done to spin up a flywheel from rest to a certain angular speed.

Torque

Definition and Calculation

• Torque (\( \tau \)): The rotational equivalent of force; causes angular acceleration.

• \( \tau = r F \sin \phi \)

• \( \tau = I \alpha \) (Newton's second law for rotation)

• \( \vec{\tau} = \vec{r} \times \vec{F} \)

Example: Calculating the torque produced by a force applied at an angle to a wrench.

Newton’s Laws for Rotation

Application

• Newton's second law for rotation: \( \sum \tau = I \alpha \)

• Combine with linear Newton's laws for systems involving both translation and rotation.

Example: Analyzing forces and torques on a pulley system.

Rotational Dynamics About a Fixed Axis

Solving Problems

• Use rotational and translational Newton's laws, kinematics, or conservation of energy as appropriate.

• Identify all forces and torques acting on the object.

Example: A disk rotating about a fixed axis with a hanging mass attached by a string.

Rotational Dynamics About a Moving Axis (Rolling Motion)

Rolling Without Slipping

• Relationship: \( v_{\text{cm}} = R \omega \)

• Use conservation of energy, including both translational and rotational kinetic energy.

Example: A solid sphere rolling down an incline without slipping.

Angular Momentum

Definition and Calculation

• For a particle: \( \vec{L} = \vec{r} \times \vec{p} \)

• For a rigid body: \( L = I \omega \)

Example: Calculating the angular momentum of a spinning disk.

Conservation of Angular Momentum

Principle

• If net external torque is zero, angular momentum is conserved: \( L_{\text{initial}} = L_{\text{final}} \)

• Often requires recalculating moment of inertia after a change (e.g., arms in/out, objects sticking together).

Example: A figure skater pulling in arms to spin faster.

Equilibrium

Conditions for Equilibrium

• Translational: \( \sum F_x = 0, \sum F_y = 0 \)

• Rotational: \( \sum \tau = 0 \)

Applications

• Plank on Supports: Analyze forces and torques to determine support reactions.

• Ladder: Consider friction, normal forces, and torques for stability.

• Strut/Beam Hanging: Include tension, weight, and wall forces.

Example: Calculating the minimum coefficient of friction to prevent a ladder from slipping.

Hooke’s Law for Stress and Strain

Elasticity in Solids

• Hooke’s Law: \( F = kx \) for springs; for solids, relates stress and strain.

• Stress: \( \text{Stress} = \frac{F}{A} \)

• Strain: \( \text{Strain} = \frac{\Delta L}{L_0} \)

• Young’s Modulus: \( Y = \frac{\text{Stress}}{\text{Strain}} \)

Example: Determining the elongation of a steel wire under a given load.

Additional info: Where the original notes were brief, standard academic context and formulas were added for completeness and clarity.