Plane motion and relative velocity

Types of plane motion

General plane motion can always be treated as a translation of a chosen base point plus a rotation about that point.

Relative velocity relation

Let A and B be two points of a rigid body having angular velocity $\omega$ (counter-clockwise positive, along $\mathbf{k}$).

        y
        |      B
        |     /
        |    /  r_B/A
        |  A
        | / r_A
        |/
        O------------ x

From the geometry, $\mathbf{r}_B = \mathbf{r}_A + \mathbf{r}_{B/A}$. Differentiating with respect to time:

$$\mathbf{v}_B = \mathbf{v}_A + \dot{\mathbf{r}}_{B/A}$$

In a rigid body the distance AB is constant, so the vector $\mathbf{r}_{B/A}$ can only rotate with angular velocity $\omega$, and $\dot{\mathbf{r}}_{B/A} = \boldsymbol{\omega}\times\mathbf{r}_{B/A}$. Hence

$$\mathbf{v}_B = \mathbf{v}_A + \boldsymbol{\omega}\times\mathbf{r}_{B/A}$$

The relative velocity $\mathbf{v}_{B/A}$ has magnitude $\omega\,(AB)$ and is perpendicular to AB, in the sense of $\omega$. To solve problems, draw the velocity triangle $\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A}$ using the known directions, or write the equation in components and solve.