Velocities of Points in Plane Motion

In plane rigid-body motion, different points generally have different velocities. Their velocities are related by the rigid-body relative-velocity equation.

Velocity relation

The velocity of any point $B$ of a plane rigid body equals the vector sum of the velocity of a selected reference point $A$ and the velocity of $B$ due to rotation about $A$:

$$\vec v_B=\vec v_A+\vec\omega\times\vec r_{B/A}.$$

The relative component is perpendicular to segment $AB$, with magnitude $v_{B/A}=\omega\,AB$.

Choosing a point $A$ with known velocity as the reference point allows the velocity of any other point $B$ to be determined. The vector $\vec\omega\times\vec r_{B/A}$ is always perpendicular to $AB$.

Velocity projections

Because the relative velocity $\vec v_{B/A}$ is perpendicular to $AB$, the velocity components of two points of a rigid body projected onto the line joining them are equal. This property often determines unknown components without a complete vector construction.

Instantaneous center of zero velocity

The instantaneous center of zero velocity (IC) is a point in the plane whose velocity is zero at the instant considered. If a finite IC $P$ exists, the velocity field of the figure at that instant is equivalent to instantaneous rotation about $P$.

Locating the IC

If the velocity directions of two points are known, draw through each point a line perpendicular to its velocity. Their intersection is the IC when the lines meet at a finite point. For pure translation, the IC is regarded as lying at infinity.

Velocity from the IC

For a point $A$ and known IC $P$, $v_A=|\omega|PA$. Thus $v_A/v_B=PA/PB$. Velocity directions are perpendicular to $PA$ and $PB$ and must correspond to one consistent sense of instantaneous rotation.

Rolling without slipping

For a wheel rolling without slip on a fixed surface, the contact point has zero instantaneous velocity and is the IC. Therefore the wheel-center speed satisfies $v_C=\omega R$ in magnitude.

Example

If the IC of a plane link is at $P$, $PA=0.2$ m, $PB=0.5$ m, and $v_A=1$ m/s, then $|\omega|=1/0.2=5$ rad/s and $v_B=5\cdot0.5=2.5$ m/s.

Common mistakes

• treating the IC as one material point fixed for a finite time interval;
• locating the IC along velocity directions instead of along perpendiculars to them;
• using $v_A/v_B=PA/PB$ without a common IC;
• assuming zero contact-point velocity when rolling includes slip.