Physics Reference Tables and Quick Lookups: Rolling Motion and Center of Mass

Abstract

This article provides a compact reference for two foundational concepts in rigid-body mechanics: the kinematic constraint for rolling without slipping and the decomposition of rigid-body motion into translation and rotation. These principles underpin the analysis of wheels, balls, and other rolling systems in classical mechanics. The article presents the key relationships, physical intuition, and worked examples suitable for quick consultation during problem-solving.

Background

Rigid-body mechanics requires understanding how objects move through space. Unlike point particles, extended objects can both translate (move their center of mass) and rotate simultaneously. Two related concepts are essential for analyzing such motion:

1. Rigid-body decomposition: Any motion of a rigid body can be expressed as translation of its center of mass plus rotation about that center ^{[center-of-mass-motion]}.
2. Rolling constraints: When an object rolls on a surface without slipping, a kinematic relationship links linear and angular velocity ^{[rolling-without-slipping]}.

These concepts are not merely theoretical—they govern the behavior of wheels, rolling balls, gyroscopes, and countless mechanical systems in engineering practice.

Key Results

Rigid-Body Motion Decomposition

The motion of a rigid body of mass $m$ can be decomposed into two independent components ^{[center-of-mass-motion]}:

• Translation: The center of mass moves with velocity $\vec{V}_{cm}$.
• Rotation: The body rotates about its center of mass with angular velocity $\vec{\omega}_{cm}$.

This decomposition is powerful because it separates the problem into two simpler subproblems. The translational motion obeys Newton's second law applied to the center of mass, while rotational motion obeys the rotational equation of motion about the center of mass.

Rolling Without Slipping Constraint

When an object rolls on a surface without slipping, the contact point between the object and surface has zero velocity relative to the surface. This imposes a kinematic constraint ^{[rolling-without-slipping]}:

$v = r\omega$

where:

• $v$ is the linear velocity of the center of mass,
• $r$ is the radius of the rolling object,
• $\omega$ is the angular velocity about the center of mass.

This relationship is not a force law but a constraint—it must be satisfied whenever rolling without slipping occurs. It ensures that the rotational and translational motions are synchronized.

Physical Interpretation

The rolling constraint emerges from the no-slip condition. Consider a point on the rim of a rolling wheel. Its velocity relative to the ground is the vector sum of:

• The velocity of the center of mass: $v$ (forward),
• The velocity due to rotation about the center of mass: $r\omega$ (tangential).

At the contact point, these must cancel for no slipping to occur, yielding $v = r\omega$.

Worked Examples

Example 1: Velocity of a Point on a Rolling Wheel

A wheel of radius $r = 0.5$ m rolls without slipping with its center moving at $v = 2$ m/s. What is the angular velocity?

Solution:

Using the rolling constraint ^{[rolling-without-slipping]}: $\omega = \frac{v}{r} = \frac{2 \text{ m/s}}{0.5 \text{ m}} = 4 \text{ rad/s}$

Example 2: Decomposing Motion of a Thrown Rod

A uniform rod of length $L$ is thrown horizontally with its center of mass moving at velocity $V_{cm} = 3$ m/s and rotating with angular velocity $\omega_{cm} = 2$ rad/s about its center.

Describe the motion.

Solution:

According to the rigid-body decomposition ^{[center-of-mass-motion]}, the rod's motion consists of:

• Translation: The center of mass travels horizontally at $V_{cm} = 3$ m/s.
• Rotation: The rod spins about its center at $\omega_{cm} = 2$ rad/s.

An observer at the center of mass sees only rotation; an observer in the lab frame sees both translation and rotation superimposed. This decomposition simplifies analysis because the translational motion (governed by the net external force) and rotational motion (governed by the net external torque about the center of mass) can be analyzed independently.

Example 3: Rolling Ball Down an Incline

A solid sphere rolls without slipping down a frictionless incline. The center of mass accelerates at $a_{cm}$. What is the angular acceleration?

Solution:

Differentiating the rolling constraint with respect to time ^{[rolling-without-slipping]}: $\frac{dv}{dt} = r \frac{d\omega}{dt}$ $a_{cm} = r \alpha$

where $\alpha$ is the angular acceleration. Thus: $\alpha = \frac{a_{cm}}{r}$

This shows that the angular acceleration is directly proportional to the linear acceleration of the center of mass, with the radius as the proportionality constant.

Summary Table

Concept	Equation	Meaning
Rolling constraint	$v = r\omega$	Linear and angular velocities are synchronized when rolling without slipping
Rolling acceleration	$a = r\alpha$	Linear and angular accelerations are synchronized
Rigid-body motion	$\vec{v}_{point} = \vec{V}_{cm} + \vec{\omega} \times \vec{r}$	Velocity of any point is translation plus rotation about center of mass

References

^{[rolling-without-slipping]}

^{[center-of-mass-motion]}

AI Disclosure

This article was drafted with the assistance of an AI language model based on personal class notes. The mathematical relationships and physical principles are drawn from cited course materials. The worked examples and summary table were generated to illustrate the concepts but have not been independently verified against external sources. Readers should cross-reference with standard mechanics textbooks (e.g., Goldstein, Kleppner & Kolenkow) before relying on this material for high-stakes applications.