#### Rolling - View 1

Rolling can be viewed as a combination of two separate motions, a purely translational motion and a purely rotational motion. Rolling involves both of these at the same time - rotation while the wheel is experiencing straight-line motion.

Let's focus on a special, but very common, rolling situation, where a wheel rolls without slipping along a horizontal surface.

#### Rolling - View 2

How does the straight-line speed of the center-of-mass compare to the rotational speed of a point on the edge of the wheel? If the wheel has a constant angular velocity, the rotational speed is:

v_{r} = 2π r / T

When a wheel rolls without slipping, the straight-line distance traveled by the wheel's center-of-mass (in red on the simulation) is exactly equal to the rotational distance traveled by a point on the edge of the wheel (in green).

Because the distances and times are equal, the translational speed of the center of the wheel equals the rotational speed of a point on the edge of the wheel.

#### Rolling - View 3

The net velocity of any point on the wheel is the vector sum of the translational velocity, which has a constant direction, and the rotational velocity, which rotates with the point as the wheel rotates.

Because the magnitudes of these two velocities are equal (let's say they're both v), the net velocity for these key points is:

top of the wheel: v + v = 2v

center of the wheel: v + 0 = v

bottom of the wheel: v - v = 0

The fact that the bottom of the wheel is instantaneously at rest makes sense - if it wasn't the wheel would be slipping as it rolled. No slipping means no relative motion where the rubber meets the road.

#### Rolling - View 4

Tracing out the path followed by a point on the edge of the wheel results in an interesting shape called a cycloid, and confirms that a point on the edge of the wheel is instantaneously at rest when it is in contact with the ground.