Rotation & Angular Dynamics · Physics

Rolling without slipping ties translation to rotation.

In pure rolling, the contact point is instantaneously at rest relative to the ground. That single constraint links v and ω, shapes the energy, and determines acceleration down an incline.

This topic

Rolling Motion Without Slipping

Learn the no-slip constraint, then use it to solve rolling problems with energy and (when needed) dynamics.

Constraint

No-slip condition (v = Rω)

Rolling without slipping imposes a geometric constraint: the center-of-mass speed and angular speed are linked by the radius.

Meaning of “no slipping” at the contact point
Constraint: v_CM = Rω
Also for accelerations: a_CM = Rα (when valid)
Common confusion: sliding vs rolling

Energy

Energy split: translation + rotation

A rolling object has both translational and rotational kinetic energy. The no-slip condition lets you write everything in terms of v or ω.

K = ½mv² + ½Iω²
Use v = Rω to combine terms
Why shape matters (I controls the split)
Energy conservation for rolling down inclines

Dynamics

Acceleration down an incline (rolling)

Rolling acceleration depends on mass distribution because torque is needed to spin the body. A larger I generally means smaller acceleration for the same slope.

Role of torque about the center
Why rolling can be slower than sliding
Qualitative dependence on I/(mR²)
When to use energy vs ΣF and Στ

Forces

Static friction’s role (and when it does work)

Static friction provides the torque that enforces rolling without slipping. In many ideal rolling problems, it does no net work on the object’s mechanical energy—yet it is essential.

Static friction can be nonzero with no slipping
Why it can do zero work in pure rolling on a rigid surface
Direction of friction depends on the situation
Common pitfall: assuming friction always opposes motion

Practice

Practice & Exercises

Practice applying the no-slip constraint, using energy conservation, and predicting how shape affects rolling outcomes.

Identify rolling vs slipping from given information
Use v = Rω and a = Rα correctly
Energy problems: solve for speed at the bottom
Incline problems: compare disk vs hoop vs sphere
Exam-style rolling motion sets