Rolling Without Slipping: Deriving the Constraint Relation

Abstract

Rolling without slipping is a fundamental kinematic constraint in mechanics that relates the linear motion of an object's center of mass to its rotational motion. This article derives the constraint relation $v = r\omega$ from first principles, explains its physical meaning, and demonstrates its application in analyzing rolling systems. The derivation clarifies why this constraint emerges naturally from the no-slip condition and how it simplifies the analysis of rolling bodies.

Background

Understanding the motion of rolling objects requires integrating two distinct types of motion: translation and rotation ^{[center-of-mass-motion]}. A rigid body in general motion can be decomposed into the translational motion of its center of mass and rotational motion about that center. When an object rolls on a surface, these two motions are not independent—they are coupled by a geometric constraint imposed by the contact condition.

The rolling-without-slipping condition specifies that the point of contact between the rolling object and the surface has zero velocity relative to the surface. This is a kinematic constraint that eliminates one degree of freedom from the system, creating a deterministic relationship between how fast the center of mass moves and how fast the object spins.

Key Results

The Constraint Relation

For an object rolling without slipping on a surface, the velocity of the center of mass $v$ and the angular velocity $\omega$ satisfy ^{[rolling-without-slipping]}:

$v = r\omega$

where $r$ is the radius of the rolling object.

Derivation

Consider a circular object of radius $r$ rolling on a flat surface. At any instant, the object rotates about its center of mass with angular velocity $\omega$. The velocity of a point on the rim due to rotation alone is $r\omega$ in the tangential direction.

The center of mass moves with velocity $v$ along the surface. From the perspective of the ground frame, the velocity of the contact point is the vector sum of:

• The translational velocity of the center of mass: $v$ (forward)
• The velocity of the contact point relative to the center of mass due to rotation: $r\omega$ (backward, since the bottom of the wheel moves opposite to the direction of rolling)

For the no-slip condition, the contact point must have zero velocity relative to the ground:

$v - r\omega = 0$

Therefore:

$v = r\omega$

This constraint is purely kinematic—it follows from geometry and the no-slip condition, independent of the forces acting on the object.

Physical Interpretation

The constraint $v = r\omega$ means that the distance traveled by the center of mass in one complete rotation equals the circumference of the object. If the object rotates through angle $\theta$, the center of mass advances by distance $r\theta$. This is the hallmark of rolling without slipping: there is no relative motion between the contact point and the surface.

This contrasts sharply with sliding, where the contact point moves relative to the surface. In pure rolling, the contact point instantaneously comes to rest, making it an instantaneous center of rotation. All points on the object rotate about this contact point at that instant ^{[center-of-mass-motion]}.

Worked Example

Problem: Velocity of a Point on a Rolling Wheel

A wheel of radius $r = 0.5$ m rolls without slipping on a horizontal surface. The center of the wheel moves at $v = 2$ m/s. Find:

1. The angular velocity of the wheel
2. The velocity of a point on the rim at the top of the wheel
3. The velocity of the contact point

Solution:

(1) Angular velocity:

Using the constraint relation: $\omega = \frac{v}{r} = \frac{2 \text{ m/s}}{0.5 \text{ m}} = 4 \text{ rad/s}$

(2) Velocity of the top point:

The top of the wheel is at distance $r$ from the center, moving in the same direction as the center. Its velocity is: $v_{\text{top}} = v + r\omega = 2 + 0.5 \times 4 = 4 \text{ m/s}$

Alternatively, the top point is at distance $2r$ from the instantaneous center of rotation (the contact point), so: $v_{\text{top}} = 2r \cdot \omega = 2 \times 0.5 \times 4 = 4 \text{ m/s}$

(3) Velocity of the contact point:

By definition of the no-slip condition: $v_{\text{contact}} = 0$

This confirms that the contact point is instantaneously at rest, serving as the instantaneous center of rotation.

References

• ^{[rolling-without-slipping]}
• ^{[center-of-mass-motion]}

AI Disclosure

This article was drafted with AI assistance. The structure, derivations, and explanations were generated based on the provided class notes, with all factual claims traced to source notes. The worked example was constructed to illustrate the constraint relation. A human expert should review the mathematical derivations and physical interpretations for accuracy before publication.