Rolling without slipping generally occurs when an object rolls without skidding. To relate this to a real world phenomenon, we can consider the example of a wheel rolling on a flat, horizontal surface.

Rolling without slipping can be better understood by breaking it down into two different motions: 1) Motion of the center of mass, with linear velocity *v* (translational motion); and 2) rotational motion around its center, with angular velocity *w*.

## Rolling Motion

Rolling motion is a combination of rotational and translational motion.

When an object is rolling on a plane without slipping, the point of contact of the object with the plane does not move. If we imagine a wheel moving forward by rolling on a plane at speed v, it must also be rotating about its axis at an angular speed

The object's angular velocity

## Rolling Without Slipping

A body rolling a distance of x on a plane without slipping.

In mathematical terms, the length of the arc is equal to the angle of the segment multiplied by the object's radius, R. Therefore, we can say that the length of the arc of the wheel that has rotated an angle θ, is equal to Rθ. Furthermore, since the wheel is in constant contact with the ground, the length of the arc correlating to the angle is also equal to x. Therefore,

Since

where