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

Rotational Motion, Dynamics, and Equilibrium

Rotational Quantities and Vector Directions

Rotational motion involves quantities analogous to linear motion, but defined for rotation about an axis. Understanding vector directions is crucial for correctly applying the right-hand rule and solving problems involving angular quantities.

• 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: Curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the angular velocity vector.

• Vector Nature: Rotational quantities are vectors and follow vector addition rules.

• Example: A wheel spinning counterclockwise (as seen from above) has an angular velocity vector pointing upward.

Moment of Inertia & Parallel Axis Theorem

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.

• Moment of Inertia (\( I \)): For discrete masses: $I = \sum M_i R_i^2$

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

• Parallel Axis Theorem: If \( I_{\text{cm}} \) is the moment of inertia about the center of mass, then about a parallel axis a distance \( d \) away: $I = I_{\text{cm}} + M d^2$

• Using Tables: Standard moments of inertia for common shapes (e.g., solid cylinder, hoop, sphere) are often provided in tables.

• Example: For a thin rod of length \( L \) and mass \( M \) about its end: $I = \frac{1}{3} M L^2$

Rotational Kinematics (Constant Angular Acceleration)

Rotational kinematics equations describe motion with constant angular acceleration, analogous to linear kinematics.

• Key Equations:

  • $\omega = \omega_0 + \alpha t$

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

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

• Variables: \( \omega \) = angular velocity, \( \alpha \) = angular acceleration, \( \theta \) = angular displacement, \( t \) = time.

• Example: A disk starts from rest and accelerates at \( 2\ \text{rad/s}^2 \) for 3 s. Find its angular velocity: $\omega = 0 + (2)(3) = 6\ \text{rad/s}$.

Rotational Kinetic Energy, Work, and Work-Energy Theorem

Rotational motion involves kinetic energy and work, analogous to linear motion but with rotational variables.

• Rotational Kinetic Energy: $K_{\text{rot}} = \frac{1}{2} I \omega^2$

• Work Done by Torque: $W = \tau \theta$ (for constant torque)

• Rotational Work-Energy Theorem: $W_{\text{net}} = \Delta K_{\text{rot}}$

• Example: A flywheel (\( I = 2\ \text{kg} \cdot \text{m}^2 \)) spinning at \( 10\ \text{rad/s} \) has $K_{\text{rot}} = \frac{1}{2} (2)(10^2) = 100\ \text{J}$.

Torque: Definitions and Calculations

Torque is the rotational equivalent of force, causing changes in rotational motion.

• Definition: $\tau = r F \sin \phi$ where \( r \) is the lever arm, \( F \) is the force, and \( \phi \) is the angle between them.

• Vector Definition: $\vec{\tau} = \vec{r} \times \vec{F}$

• Units: Newton-meter (N·m)

• Example: A 5 N force applied 0.2 m from the axis at 90°: $\tau = (0.2)(5) = 1\ \text{N} \cdot \text{m}$.

Newton’s Laws for Rotation

Newton's second law for rotation relates net torque to angular acceleration.

• Rotational Form: $\sum \tau = I \alpha$

• Application: Analyze all torques acting on a body to solve for unknowns.

• Example: A disk (\( I = 0.5\ \text{kg} \cdot \text{m}^2 \)) under a net torque of 2 N·m: $\alpha = \frac{2}{0.5} = 4\ \text{rad/s}^2$.

Rotational Dynamics About a Fixed Axis

Problems may require combining rotational and translational equations or using energy methods.

• Fixed Axis: Axis does not move; use $\sum \tau = I \alpha$ and kinematic equations.

• Energy Approach: Use conservation of energy if non-conservative forces are absent.

• Example: A pulley system where a mass falls, causing rotation—analyze using both force and torque equations.

Rotational Dynamics About a Moving Axis (Rolling Motion)

Rolling motion involves both rotation and translation. The relationship between linear and angular variables is essential.

• Rolling Without Slipping: $v_{\text{cm}} = R \omega$

• Total Kinetic Energy: $K = \frac{1}{2} M v_{\text{cm}}^2 + \frac{1}{2} I_{\text{cm}} \omega^2$

• Conservation of Energy: Often used to solve rolling problems.

• Example: A solid sphere rolls down an incline; use energy conservation to find its speed at the bottom.

Angular Momentum: Definitions and Calculations

Angular momentum is a measure of rotational motion, conserved in isolated systems.

• Definition for a Particle: $\vec{L} = \vec{r} \times \vec{p}$

• For a Rigid Body: $L = I \omega$

• Units: kg·m2/s

• Example: A disk (\( I = 0.2\ \text{kg} \cdot \text{m}^2 \)) spinning at 5 rad/s: $L = (0.2)(5) = 1\ \text{kg} \cdot \text{m}^2/\text{s}$.

Conservation of Angular Momentum

In the absence of external torques, the total angular momentum of a system remains constant.

• Conservation Law: $L_{\text{initial}} = L_{\text{final}}$

• Application: Used in problems involving changing moments of inertia (e.g., figure skater pulling in arms).

• Example: A rotating platform with a student who moves inward, decreasing \( I \) and increasing \( \omega \) to conserve \( L \).

Equilibrium: Plank, Ladder, Strut, or Beam

Equilibrium problems involve analyzing forces and torques to ensure an object remains at rest.

• Conditions for Equilibrium:

  • $\sum F_x = 0$ (no net force in x-direction)

  • $\sum F_y = 0$ (no net force in y-direction)

  • $\sum \tau = 0$ (no net torque about any axis)

• Applications: Plank on supports, ladder against a wall, beams under tension.

• Example: A uniform ladder leaning against a wall—analyze forces at the base and wall, and torques about the base.

Hooke’s Law: Stress and Strain

Hooke’s law describes the relationship between force and deformation for elastic objects under tension or compression.

• Hooke’s Law: $F = k \Delta x$ (for springs)

• Stress: $\text{Stress} = \frac{F}{A}$ (force per unit area)

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

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

• Example: A steel wire (\( A = 1\ \text{mm}^2 \), \( L_0 = 2\ \text{m} \)) stretches 1 mm under a 100 N load. Calculate stress, strain, and Young’s modulus.

Summary Table: Key Rotational Quantities

Quantity	Symbol	SI Unit	Equation
Angular Displacement	\( \theta \)	radian (rad)	—
Angular Velocity	\( \omega \)	rad/s	$\omega = \frac{d\theta}{dt}$
Angular Acceleration	\( \alpha \)	rad/s2	$\alpha = \frac{d\omega}{dt}$
Moment of Inertia	\( I \)	kg·m2	$I = \sum M_i R_i^2$
Torque	\( \tau \)	N·m	$\tau = r F \sin \phi$
Angular Momentum	\( L \)	kg·m2/s	$L = I \omega$

Additional info: Academic context and examples have been added to expand on the brief points in the original study guide, ensuring the notes are self-contained and suitable for exam preparation.