Dynamics

General Plane Motion Is Translation Plus Rotation

Translation+rotation split, rolling v=rω (d=rθ), point velocity v_P=v_O+ω×r (bottom 0, top 2v), instant center IC, cycloid

A rolling wheel slides sideways (translation) while it spins (rotation) at the same time. Every planar motion of a rigid body splits this way — translation of one point plus rotation about it — and if it does not slip, the two are tied together by v=rω.

Roll the wheel. The center slides in a straight line (translation) and the spoke turns (rotation). Their sum is general plane motion — whatever reference point you pick, the motion splits into translation plus rotation.

In rolling without slipping, the rim arc that touched the ground equals the distance traveled, so d = rθ. Differentiate in time and the center speed and angular speed lock together as v = rω.

The velocity of a point on the wheel is the center's translation plus the rotation. The contact point gives v − rω = 0 (instantly at rest), the center is v, and the top is fastest at v + rω = 2v.

Since the contact point has zero velocity, at that instant the whole wheel turns about it. That point is the instantaneous center of rotation (IC): every velocity is perpendicular to its line from the IC, with magnitude ω×(distance to the IC).

Follow one rim point over time and it traces a curve called a cycloid. Each time it touches the ground its speed is 0, making a sharp cusp — precisely because that point is the instantaneous center.

In PracticeTo sum up, the general plane motion of a rigid body can always be split into translation + rotation. Add the translation of one reference point to the rotation about it (ω×r) and you get the velocity of any point — v_P = v_O + ω×r_P/O. For rolling without slipping, the condition that the contact point has zero velocity gives v = rω, and the acceleration follows from one more derivative as a = rα plus the centripetal term rω². Taking the zero-velocity contact point as the instantaneous center of rotation (IC) lets you read every point's velocity from a single rotation, v = ω·d, which is why bottom 0, center v, top 2v jump out at once. The cusp of the cycloid traced by a rim point tells the same story. In the next lesson we generalize this v_P = v_O + ω×r relation into relative motion in a rotating frame, solving the velocities and accelerations of bodies turning relative to one another (linkages, gears, and the like).

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