Explain general plane motion with example

General plane motion (GPM) is the motion of a rigid body in which every particle stays in a plane parallel to a fixed plane, and the motion is neither pure translation nor pure rotation about a fixed axis. It is the sum of (i) a translation of a chosen reference point and (ii) a rotation of the body about that point.

For two points $A$ and $B$ of the same rigid body:

$$\vec v_B = \vec v_A + \vec\omega \times \vec r_{B/A}, \qquad \vec a_B = \vec a_A + \vec\alpha \times \vec r_{B/A} - \omega^2 \vec r_{B/A}$$

Here $v_{B/A} = \omega\, r_{B/A}$ is perpendicular to $AB$, and $a_{B/A}$ has a tangential part $\alpha r$ and a normal part $\omega^2 r$ towards $A$.

Examples

• A wheel rolling on the road: the centre translates while the wheel rotates about the centre.
• The connecting rod of an engine: end $A$ (crank pin) moves on a circle, end $B$ (piston) moves in a straight line.
• A ladder sliding down a wall.

Worked example. For a wheel of radius $r$ rolling without slipping with centre speed $v$: the centre $O$ translates with $v$ and the wheel rotates with $\omega = v/r$. The top point has $v_{top} = v + \omega r = 2v$ and the contact point has $v_{contact} = v - \omega r = 0$, so the contact point is the instantaneous centre.

        top: 2v ->
         .---.
   v ->  | O |
         '---'
     contact point: v = 0