Rolling

Rolling without slipping as a superposition of translational and rotational motion.

General idea of the topic

Rolling without slipping is a combination of translation and spin in which the contact point with the ground is instantaneously at rest. This imposes a simple constraint: the velocity of the center is v = ωR and the acceleration is a = αR.

The topic introduces the condition, analyzes the dynamics of the body rolling without slipping, explains why it serves as a bridge between translation, rotation and energy, and proposes a strategy for solving problems, starting by deciding the regime.

What you will truly understand

By the end you will be able to write the rolling constraint, compute the acceleration of cylinders and spheres rolling on a plane and choose between force and moment equations or an energy balance.

You will also be able to identify when rolling is not possible and the body slips.

Recommended learning route

1. No-Slip Rolling Condition: relation between the velocity of the center, the angular velocity and the radius, and the contact point at rest.
2. Rolling Without Slipping: dynamics of a body rolling without slipping and standard examples.
3. Rolling as a Bridge: rolling as a bridge between translation, rotation and energy.
4. Strategy for Rolling Problems: steps for solving problems: regime, constraint, equations and check.

Physical criterion you must acquire

The essential criterion is that rolling without slipping does not dissipate energy through static friction: the friction exists, but its point of application does not move. The acceleration of a body rolling down a plane depends on its shape through I/(mR²), not on its mass or radius.

Connection with real problems

• vehicle and bicycle wheels.
• cylinders and spheres on ramps.
• bearings.
• ball sports.

Final goal of the topic

The final goal is to analyze any rolling problem rigorously, justify the constraint and obtain accelerations, forces and speeds.