Rolling Wheel Velocity Field

A wheel rolling without slipping combines two motions:

1. Translation of the center with speed $v_0$ to the right.
2. Rotation about the center with angular speed $\omega = \dfrac{v_0}{R}$.

For a point $A$ on the rim, let $\varphi$ be the angle measured from the top vertical radius to the radius $OA$. Then the point’s velocity is the vector sum of the translational velocity and the tangential rotational velocity.

Components

Using the geometry of circular motion, the velocity components are:

$$v_x = v_0(1+\cos\varphi), \qquad v_y = -v_0\sin\varphi$$

where the negative sign means downward on the right side of the wheel and upward on the left side, depending on $\varphi$.

Speed

The magnitude of the total velocity is

$$v = \sqrt{v_x^2+v_y^2} = 2v_0\cos\frac{\varphi}{2}$$

for points on the upper half of the wheel, and more generally the vector form is obtained from the sum of the two motion contributions.

Direction of velocity

The total velocity of point $A$ is always perpendicular to the line $BA$, where $B$ is the instantaneous contact point with the road. This is a hallmark of instantaneous rotation about the contact point.

Special points on the rim

• Top point: speed $2v_0$
• Contact point: speed $0$
• Side points: speed $\sqrt{2}\,v_0$

Graph on the vertical diameter

For the current position, the vertical diameter contains points whose speeds vary smoothly from zero at the bottom contact point to $2v_0$ at the top point. The visualization shows this as a moving speed profile along the diameter, together with the wheel, velocity vectors, and the instantaneous center of rotation.

Use the sliders to change the wheel speed, radius, and the angle $\varphi$ of the selected rim point.