Rolling Without Slipping

Combined translational and rotational motion without relative sliding.

General idea of the topic

In rolling without slipping, static friction acts at the contact point and couples translation and spin. The second law is written for the center of mass, the rotation law about it and the constraint a = αR, and the system is solved for the acceleration and the friction.

The topic develops that dynamic analysis and applies it to the standard cases: solid cylinder, hoop, sphere and disk, both on a horizontal plane with an applied force and on an incline.

What you will truly understand

By the end you will be able to compute the acceleration of each shape when rolling, the friction force that makes it possible and the maximum angle of inclination for no slipping.

You will also be able to rank the bodies by their acceleration on a plane according to I/(mR²).

Recommended learning route

1. Dynamic Analysis of Rolling: setup and solution of the dynamics of a body rolling without slipping.
2. Standard Rolling Examples: examples of cylinder, hoop, sphere and disk and comparison of results.

Physical criterion you must acquire

The essential criterion is that the required friction must not exceed μN: otherwise the body slips. On an incline, a = g sin θ/(1 + I/(mR²)), so the smaller I/(mR²), the larger the acceleration. This factor must give the acceleration of a frictionless block when I tends to zero.

Connection with real problems

• wheels and rollers.
• ramps and races of rolling objects.
• brakes and vehicle starts.
• design of lightweight wheels.

Final goal of the topic

The final goal is to solve and compare the rolling of any rigid body on a plane, justifying the absence of slipping.