Angular Velocity (ω = θ/t)

Angular velocity is the rotational twin of ordinary speed. Where speed counts metres covered per second, angular velocity counts angle swept per second: $\omega = \theta/t$. The reason it deserves its own quantity is that every point on a rigid spinning body shares the same $\omega$ while having a completely different linear speed — the hub of a wheel and the tread of the tyre turn through the same angle in the same second, which is exactly what makes the body rigid.

A record turntable at 33⅓ rpm turns through 33.33 revolutions each minute. Convert properly: $33.33 \times 2\pi = 209.4$ rad per minute, divided by 60 s gives $\omega \approx 3.49$ rad/s. A four-pole induction motor on 60 Hz mains runs near 1750 rpm, which is about 183 rad/s. A minute hand on a clock manages $2\pi/3600 \approx 0.00175$ rad/s.

Radians are the natural currency here, and it is worth knowing why rather than just accepting it. A radian is defined as the angle that subtends an arc equal to the radius, so arc length is $s = r\theta$ with no constant attached — but only when $\theta$ is in radians. Everything downstream inherits that cleanliness: $v = \omega r$, $a_c = \omega^2 r$, $\tau = I\alpha$, $P = \tau\omega$ all come out with no conversion factor at all. Use degrees and every one of them needs a $\pi/180$ glued on, which is precisely the kind of thing that gets forgotten.

Feeding rev/min into an equation that expects rad/s is the single most common error in rotational mechanics, and it is worth memorising the conversion once: multiply rpm by $2\pi/60$, which is 0.10472. Do it the other way and multiply by 9.5493. The error is a factor of 9.55, and because it is not a round number it does not announce itself the way a factor of ten would — the answer just comes out wrong and plausible. It gets worse downstream: anything with $\omega^2$ in it, like rotational kinetic energy or centripetal acceleration, is then wrong by a factor of 91. A related slip is treating one revolution as 360 in this formula; in radians one revolution is $2\pi \approx 6.283$, and mixing the two is a factor of 57.3. This page's unit selectors will do the conversion for you, but the habit of checking which unit a supplier's data sheet used is worth more than any calculator.