Rotation III: Rolling Motion, Rotational Kinetics, and Angular Momentum

Rotation III: Rolling Motion, Rotational Kinetics, and Angular Momentum

Overview

This study guide covers advanced topics in rotational motion, including rolling without slipping, rotational kinetic energy, and angular momentum. These concepts are essential for understanding the dynamics of rotating bodies in physics, with applications ranging from mechanical systems to astrophysical phenomena.

Rolling Without Slipping

Definition and Conditions

Rolling without slipping occurs when an object rolls on a surface such that the point of contact is momentarily at rest relative to the surface. This is a common scenario for wheels, cylinders, and spheres in motion.

• Rolling Constraint: The velocity of the center of mass is related to the angular velocity by $v_{CM} = \omega R$.

• Condition for No Slipping: The static friction force is sufficient to prevent relative motion at the contact point.

• Period and Displacement: For one full rotation, the object travels a distance equal to its circumference: $\Delta x = 2\pi R$.

Example: A wheel of radius $R$ rolling without slipping has its center of mass move at $v_{CM}$, while the point at the top moves at $2v_{CM}$ and the bottom at $0$.

Rotational Kinetic Energy

Energy of Rotating Bodies

Rotational kinetic energy is the energy due to the rotation of an object and is analogous to translational kinetic energy.

• Formula: $KE_{rot} = \frac{1}{2} I \omega^2$

• Total Kinetic Energy (Rolling Object): $KE_{total} = \frac{1}{2} m v_{CM}^2 + \frac{1}{2} I \omega^2$

• Moment of Inertia ($I$): Depends on mass distribution. See table below.

Example: For a solid disk of mass $M$ and radius $R$, $I = \frac{1}{2}MR^2$.

Object	Axis	Moment of Inertia ($I$)
Hoop	About cylinder axis	$MR^2$
Solid cylinder/disk	About cylinder axis	$\frac{1}{2}MR^2$
Thin rod	Axis through center, length $L$	$\frac{1}{12}ML^2$
Solid sphere	Any diameter	$\frac{2}{5}MR^2$
Hoop	Any diameter	$\frac{1}{2}MR^2$

Translational and Rotational Dynamics

Combining Forces and Torques

When analyzing systems with both translation and rotation, Newton's second law and the rotational analog must be applied together.

• Translational Motion: $\sum \vec{F} = m \vec{a}$

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

• Strategy: Draw free-body diagrams, identify forces and torques, and apply both laws as needed.

Example: A bucket attached to a string wound around a cylinder involves both the translation of the bucket and the rotation of the cylinder.

Sample Problems and Applications

Problem 1: Angular Displacement of a Race Car Tire

Given: Tire radius $r = 0.33$ m, average speed $v_{CM} = 83$ m/s, time $\Delta t = 1.5$ hours.

• Angular Displacement: $\Delta \theta = \omega \Delta t = \frac{v_{CM}}{r} \Delta t$

• Calculation: $\Delta \theta = \frac{83}{0.33} \times 5400 \approx 1.3 \times 10^6$ radians

Problem 2: Work and Power to Stop a Rotating Wheel

Given: Mass $m = 32.0$ kg, radius $r = 1.20$ m, initial angular velocity $\omega_i = 28$ rad/s, time $\Delta t = 15.0$ s.

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

• Work Done: Equal to the initial kinetic energy, since final $\omega = 0$.

• Average Power: $P_{avg} = \frac{W}{\Delta t}$

Problem 3: Neutron Star Collapse

Given: Initial radius $R_i = 7 \times 10^5$ km, final radius $R_f = 100$ km, initial period $T_i = 10$ days.

• Conservation of Angular Momentum: $I_i \omega_i = I_f \omega_f$

• Moment of Inertia for Solid Sphere: $I = \frac{2}{5}MR^2$

• Final Angular Speed: $\omega_f = \omega_i \frac{I_i}{I_f}$

Problem 4: Rotational Collision (Disk and Loop)

Given: Solid disk and loop, both 1.0 kg, diameter 20.0 cm, initial disk rotation 200 rpm.

• Conservation of Angular Momentum: $I_1 \omega_1 = (I_1 + I_2) \omega_f$

• Final Angular Velocity: $\omega_f = \frac{I_1 \omega_1}{I_1 + I_2}$

Angular Momentum

Definition and Conservation

Angular momentum is a measure of the rotational motion of an object and is conserved in the absence of external torques.

• Formula: $\vec{L} = I \vec{\omega}$

• Conservation Law: $\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$; if $\sum \vec{\tau}_{ext} = 0$, then $\vec{L}$ is constant.

• Applications: Used in analyzing collisions, star collapse, and rotational systems.

Example: When a spinning ice skater pulls in their arms, their moment of inertia decreases and angular velocity increases to conserve angular momentum.

Summary Table: Rotational Inertia of Common Objects

Object	Moment of Inertia ($I$)
Thin Hoop (axis through center)	$MR^2$
Solid Cylinder/Disk (axis through center)	$\frac{1}{2}MR^2$
Solid Sphere (diameter)	$\frac{2}{5}MR^2$
Thin Rod (center)	$\frac{1}{12}ML^2$

Key Equations

• $v_{CM} = \omega R$

• $KE_{total} = \frac{1}{2} m v_{CM}^2 + \frac{1}{2} I \omega^2$

• $\sum \vec{F} = m \vec{a}$

• $\sum \tau = I \alpha$

• $\vec{L} = I \vec{\omega}$

• $\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$

Additional info: These notes expand on brief lecture points and diagrams, providing full definitions, formulas, and context for college-level physics students studying rotational dynamics.