Rolling Without Slipping: Constraint, Forces, and a Solid-Sphere Example

Rolling without slipping is a kinematic condition: the point of contact has zero instantaneous velocity relative to the surface. For a circular body of radius $R$ on a stationary surface, this gives $v_{cm}=R\omega$ and, while the contact remains in the no-slip regime, $a_{cm}=R\alpha$ in magnitude. On an incline, a solid sphere released from rest rolls without slipping only when static friction can supply the required torque; its acceleration is then $a_{cm}=\frac{5}{7}g\sin\theta$. On a truly frictionless incline, it does not begin rolling: it slides with $a_{cm}=g\sin\theta$ and has no angular acceleration from the incline.

The no-slip constraint

Rigid-body motion can be described as translation of the center of mass plus rotation about the center of mass. At the contact point, the translational and rotational velocities cancel during pure rolling. The relation $v_{cm}=R\omega$ is therefore a constraint between the two motions, not a force law and not a guarantee that a body will roll without slipping. For motion down a straight incline, differentiating the constraint gives $a_{cm}=R\alpha$ in magnitude. OpenStax derives these contact-point relations and distinguishes rolling from slipping.

Static friction is what can enforce this constraint when a body accelerates down an incline. It acts at the contact point and provides torque about the center of mass. Its magnitude adjusts as needed up to $f_s\leq\mu_sN$; if the required force exceeds that limit, the body slips and the no-slip equations no longer apply.

Example: a solid sphere on an incline

Consider a uniform solid sphere released from rest on an incline of angle $\theta$. Assume a rigid sphere, a fixed plane, negligible rolling resistance and air drag, and enough static friction to prevent slipping. Let positive translation point down the plane and let $f_s$ denote the magnitude of static friction, which points up the plane.

Translation along the plane gives

$mg\sin\theta-f_s=ma.$

Taking torques about the center of mass and using $a=R\alpha$ gives

$f_sR=I\alpha=\frac{2}{5}mR^2\frac{a}{R},$

so $f_s=\frac{2}{5}ma$. Substituting into the force equation,

$a=\frac{5}{7}g\sin\theta,\qquad f_s=\frac{2}{7}mg\sin\theta.$

Since $N=mg\cos\theta$, the no-slip condition requires

$\mu_s\geq\frac{f_s}{N}=\frac{2}{7}\tan\theta.$

If the surface cannot provide that much static friction, these results do not describe the motion; the sphere slips and its translation and rotation must be solved with the appropriate friction model. The rotational-inertia dependence is general: for an object with center-of-mass moment of inertia $I$, pure rolling down the incline gives $a=g\sin\theta/(1+I/(mR^2))$, provided static friction is sufficient. OpenStax presents this force-and-torque method and the corresponding friction requirement for rolling objects.

What changes on a frictionless incline?

If the incline is frictionless, there is no tangential contact force and therefore no torque about the sphere's center from the plane. A sphere released from rest then has $a=g\sin\theta$ and remains non-rotating in the ideal model. The value $\frac{5}{7}g\sin\theta$ belongs to a solid sphere rolling without slipping with sufficient static friction; it is not the answer for a frictionless incline.

In ideal pure rolling on a stationary rigid surface, static friction can be nonzero while doing zero instantaneous work at the stationary contact point. This idealization does not mean real rolling has no energy losses: deformation and other effects produce rolling resistance. Static friction, kinetic friction during slipping, and rolling resistance describe different parts of the model.

References

• Moebs, W., Ling, S. J., & Sanny, J. (2016). 11.1 Rolling Motion, University Physics Volume 1. OpenStax. Used for the no-slip kinematics, incline force/torque method, static-friction condition, and ideal static-friction work discussion.
• MIT OpenCourseWare. 36.2 Worked Example: Wheel Rolling Without Slipping Down an Inclined Plane (Torque Method), Classical Mechanics. Supplemental worked example using the same force-and-torque approach.

AI Disclosure

This article was originally drafted with AI assistance from personal class notes. The worked example and explanatory text in this revision were checked against the primary textbook and lecture references above. The equations assume the ideal rigid-body, no-slip model stated in the text.