general plane motion rigid body
A rolling wheel moves sideways (translation) and turns in place (rotation) at the same time. However complicated it looks, any motion of a rigid body in a plane can be split this way into the translation of one point plus a rotation about that point. If the wheel rolls without slipping, the two motions are not independent but locked together by v=rω. As a result the points on the wheel all have different velocities: the point touching the ground is momentarily at rest and the top is the fastest. This lesson combines translation and rotation to read off the velocity of any point on a wheel.
Roll the wheel with the slider and its center moves along a straight line parallel to the ground while the gold spoke turns about that center. The dashed circle is the wheel at the start, the stretch labeled in blue is the translation of the center, and the orange arc is the rotation over that time. Adding the two gives general plane motion, and whichever point you take as the reference, the motion splits into that point's translation and a rotation about it. Changing the reference point changes the translational part, but the rotational part, the angular velocity, is the same seen from any point, which is what matters.
When a wheel rolls without slipping, the arc of the rim that has touched the ground equals exactly the distance the wheel has rolled. Roll the wheel with the slider and the green line painted on the ground and the green arc around the wheel always grow to the same length together. So d = rθ holds, and differentiating both sides with respect to time ties the velocity of the center to the angular velocity: v = rω. A wheel spinning in place would make the arc longer than the distance, and one dragged along skidding would make it shorter, so this equation is precisely the no-slip condition.
The velocity of a point on the wheel is the translational velocity of the center plus the rotational velocity of that point. At the contact point with the ground the rotational velocity is exactly opposite the translational one, so v − rω = 0 and the point is momentarily at rest; the center moves at v, its translational velocity alone. At the top the two velocities point the same way, v + rω = 2v, the fastest of all. Change the speed with the slider and the three arrows grow and shrink together, always in the ratio 0 to 1 to 2. That speed difference is why the upper spokes of a bicycle wheel look blurred while the lower ones look sharp.
Since the contact point has velocity 0, at that instant the whole wheel is effectively turning about the contact point. This point is called the instantaneous center of rotation (IC), and the velocity of any point on the wheel is perpendicular to its IC line, the dashed line joining it to the contact point, with magnitude ω × (distance to the IC). Change the angular velocity with the slider and all the arrows grow together, longer the farther a point is from the contact, so the top is the fastest. It is a way to read every point's velocity at a glance from a single rotation, without splitting the motion into translation and rotation.
Follow one point on the rim of the wheel and it traces a curve called a cycloid. Increase the rolling angle φ with the slider and the gold point climbs and falls along an arch, with its height shown in the readout. Each time the point touches the ground its velocity is 0 and the curve forms a sharp cusp, because that spot is the instantaneous center of rotation. The width of one arch equals the distance the wheel rolls in one turn, its circumference.