Part II: Dynamics · Topic 7 of 8 · 22 min read

Rigid-Body Kinematics

Track one point, one rotation and geometric constraints.

What you'll learn

Use relative-velocity vectors to analyse rigid planar linkages.
Construct correct tangential and normal relative-acceleration terms.
Apply slider/guide constraints without relying on an instantaneous centre.
Use rolling without slip, and distinguish its velocity and acceleration consequences.

A straight rigid bar inclined about 22 degrees, from pin A at lower left to pin B at upper right. A vertical guide line runs through A and a horizontal dashed guide line runs through B. A dark blue arrow points up from A. A dark red arrow points right from B. Labels: A; B; vA; vB. Centred caption below: Relative velocity of B with respect to A ⟂ AB

Understand the concepts

All points share one angular velocity

For a planar rigid body, the velocities of points $A$ and $B$ differ by an angular motion about their relative position. Relative velocity $\mathbf v_B-\mathbf v_A$ is perpendicular to the line .

Assumptions and sign conventions

Planar rigid body with no deformation.
A, B position vectors must refer to the same instant.
Use $+x$ right, $+y$ up, and CCW angular motion positive.
For a slider constrained to a straight stationary guide, normal coordinate velocity and acceleration vanish.

Equations that matter

\mathbf v_B=\mathbf v_A+\boldsymbol\omega\times\mathbf r_{B/A}

Step-by-step method

Draw a position diagram and label all instantaneous dimensions and motion constraints.
Choose two points on the same rigid body and form $\mathbf r_{B/A}$ .
Write the full relative-velocity vector equation; project onto x and y.
For acceleration use the already determined angular velocity to compute the $-\omega^2\mathbf r$ term.

Worked example: Rigid bar between perpendicular sliders

A rigid bar joins slider $A$ at $(0,0)$ and slider $B$ at $(0.40,0.30)\,\mathrm m$ . moves vertically upward with and has . Slider is restricted to a horizontal line. Determine , , and .

Common mistakes

Using the instantaneous centre to compute acceleration as if it were a fixed pin.
Assuming relative acceleration is only perpendicular to a rotating rod.
Applying relative velocity between points that are not on the same rigid body.
Using the wrong direction for normal relative acceleration (it points from B toward A).

Check yourself

Is relative velocity between two rigidly linked points parallel or perpendicular to the link?
Does a point with zero instantaneous velocity necessarily have zero acceleration?
Which term survives when angular velocity is constant but nonzero?

Ready to try it yourself?

Try Practice Questions (4)

Related: Particle Kinematics

Independent Waypoint Mentor learning material. Not affiliated with or endorsed by any university.

Rigid-Body Kinematics

What you'll learn

Understand the concepts

All points share one angular velocity

Assumptions and sign conventions

Equations that matter

Step-by-step method

Worked example: Rigid bar between perpendicular sliders

Common mistakes

Check yourself

Acceleration has two distinct relative parts

The instantaneous centre is a velocity shortcut

Geometric constraints are equations

Rolling contact needs care

Step 1: Write the position vector

Step 2: Use relative velocity

Step 3: Find angular speed and B velocity

Step 4: Use relative acceleration

Step 5: Solve both components