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 objects rotating about an axis. Understanding the direction of these vectors is crucial for solving problems in rotational dynamics.

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

• Angular Velocity (\( \vec{\omega} \)): The rate of change of angular displacement. Direction is given by the right-hand rule.

• Angular Acceleration (\( \vec{\alpha} \)): 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.

• Example: A wheel rotating counterclockwise in the plane of the page has an angular velocity vector pointing out of the page.

Moment of Inertia & Parallel Axis Theorem

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

• Discrete Masses: \( I = \sum M_i R_i^2 \), where \( M_i \) is the mass of the i-th particle and \( R_i \) is its distance from the axis.

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

• 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: $I = I_{\text{cm}} + Md^2$

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

• Example: The moment of inertia of a thin rod of mass \( M \) and length \( L \) about its center: $I_{\text{center}} = \frac{1}{12}ML^2$

Rotational Kinematics (Constant Angular Acceleration)

Rotational kinematics describes the motion of rotating objects under constant angular acceleration, analogous to linear kinematics.

• Key Equations:

  • $\omega = \omega_0 + \alpha t$

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

  • $\omega^2 = \omega_0^2 + 2\alpha \theta$

• Variables: \( \omega \) = final angular velocity, \( \omega_0 \) = initial angular velocity, \( \alpha \) = angular acceleration, \( t \) = time, \( \theta \) = angular displacement.

• 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, Rotational Work, and the Work-Energy Theorem

Rotating objects possess kinetic energy due to their motion. Work done by torques changes this energy.

• 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: The net work done by torques equals the change in rotational kinetic energy: $W_{\text{net}} = \Delta K_{\text{rot}}$

• Example: A flywheel (\( I = 2\ \text{kg} \cdot \text{m}^2 \)) spins up from rest to \( 10\ \text{rad/s} \): $K_{\text{rot}} = \frac{1}{2}(2)(10^2) = 100\ \text{J}$

Torque: Definitions and Calculations

Torque (\( \tau \)) is the rotational equivalent of force, causing angular acceleration.

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

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

• Multiple Forces: Total torque is the sum of individual torques.

• Example: A 10 N force applied 0.5 m from the axis at 90°: $\tau = (0.5)(10)\sin 90^\circ = 5\ \text{N} \cdot \text{m}$

Newton’s Laws for Rotation

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

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

• Application: Used to solve for unknowns in rotational systems, such as angular acceleration or required torque.

• Example: A disk (\( I = 0.2\ \text{kg} \cdot \text{m}^2 \)) experiences a net torque of 1 N·m: $\alpha = \frac{1}{0.2} = 5\ \text{rad/s}^2$

Rotational Dynamics About a Fixed Axis

When an object rotates about a fixed axis, both rotational and translational equations may be needed, along with kinematics or energy conservation.

• Combined Analysis: Use $\sum \tau = I\alpha$ and $F = ma$ as needed.

• Energy Approach: $K_{\text{rot}} + K_{\text{trans}} + U = \text{constant}$ (if no non-conservative forces)

• Example: A pulley system where a mass falls, causing a disk to rotate. Use both rotational and linear equations to solve for acceleration.

Rotational Dynamics About a Moving Axis (Rolling Motion)

Rolling motion involves both rotation and translation, such as a wheel rolling without slipping.

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

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

• Energy Conservation: Often used to solve for speed or height in rolling problems.

• Example: A solid sphere rolls down an incline. Use energy conservation and $v_{\text{cm}} = R\omega$ to find final speed.

Angular Momentum: Definitions and Calculations

Angular momentum (\( L \)) is a measure of rotational motion, analogous to linear momentum.

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

• For a Rigid Body: $L = I\omega$ (about a fixed axis)

• Direction: Given by the right-hand rule.

• Example: A disk (\( I = 0.5\ \text{kg} \cdot \text{m}^2 \)) spinning at 4 rad/s: $L = (0.5)(4) = 2\ \text{kg} \cdot \text{m}^2/\text{s}$

Conservation of Angular Momentum

Angular momentum is conserved if the net external torque on a system is zero.

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

• Application: Used in problems involving collisions or changes in moment of inertia.

• Example: A figure skater pulls in her arms, reducing \( I \) and increasing \( \omega \) so that $I_1\omega_1 = I_2\omega_2$

Equilibrium: Plank, Ladder, Strut, or Beam

Equilibrium occurs when the net force and net torque on an object are zero. This is essential for analyzing static structures.

• Conditions for Equilibrium:

  • $\sum F_x = 0$

  • $\sum F_y = 0$

  • $\sum \tau = 0$

• Applications: Planks on supports, ladders against walls, beams under tension.

• Example: A uniform plank rests on two supports. Set up force and torque equations to solve for support forces.

Hooke’s Law: Stress and Strain

Hooke’s law describes the relationship between the force applied to an object and its resulting deformation, within the elastic limit.

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

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

• Strain: $\text{Strain} = \frac{\Delta L}{L_0}$ (fractional change in length)

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

• Example: A steel wire (\( Y = 2 \times 10^{11} \ \text{N/m}^2 \)) stretches by 1 mm under a 1000 N load. Use the formulas above to find the required diameter.

Summary Table: Key Rotational Quantities

Quantity	Symbol	SI Unit	Formula
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	$\tau = rF\sin\phi$
Rotational Kinetic Energy	\( K_{\text{rot}} \)	J	$K_{\text{rot}} = \frac{1}{2}I\omega^2$
Angular Momentum	\( L \)	kg·m2/s	$L = I\omega$

Additional info: Some explanations and examples have been expanded for clarity and completeness, as the original study guide was in outline form.