Linear Speed from Rotation (v = ωr)

Every point on a rigid spinning body shares the same angular velocity, but the farther a point sits from the axis, the faster it actually travels through space. The relationship is exactly linear in the radius: $v = \omega r$. It comes straight from the definition of the radian — arc length is $s = r\theta$, so dividing both sides by time gives $v = r\omega$, with no constant to remember. This is why the outer horses on a carousel feel so much livelier than the inner ones, and why the difference between the two is precisely the ratio of their radii.

A 300 mm circular saw blade at 3600 rpm: convert first, $\omega = 3600 \times 0.10472 = 377$ rad/s, and the radius is 0.15 m. The rim speed is $v = 377 \times 0.15 \approx 57$ m/s, over 200 km/h. That number is not academic — it is the figure a blade's maximum-rpm rating is derived from, and it is why fitting a larger blade to a faster saw is dangerous even when it physically fits.

The relation is the whole basis of gearing and belt drives. Two pulleys joined by a belt share a common belt speed $v$, so $\omega_1 r_1 = \omega_2 r_2$ and the speed ratio is the inverse of the radius ratio. Meshing gears share a common pitch-line velocity and behave identically. In machining it appears as surface speed, the quantity that actually governs tool wear, which is why a lathe's constant-surface-speed mode continuously changes the spindle rpm as the tool moves across a face.

The measurement that goes wrong is which radius. Component data is quoted in diameters far more often than radii — a "300 mm blade", a "600 mm sheave", a "50 mm bar" — and entering the diameter doubles the answer. Worse, on a V-belt drive the correct figure is the pitch diameter where the belt actually sits in the groove, not the outside diameter of the sheave, and the two can differ by 10 mm or more on a small pulley; on a gear it is the pitch diameter, not the tip diameter. A speed ratio computed from outside diameters is close enough to look right and wrong enough to matter. The other error is the familiar one of feeding rpm straight in where the equation wants rad/s, which multiplies the answer by 9.55 — a 3600 rpm blade would appear to have a rim speed of 540 m/s, which is faster than sound and ought to prompt a second look.