Understanding angular velocity
rad/s, rpm, hertz — rotation, three notations.
The three units of rotational speed, the 2π that bridges them, and where each is conventional.
The three units.
Radians per second (rad/s) is the SI unit. Revolutions per minute (rpm) is the engineering / automotive / motor unit. Hertz (Hz) is the frequency unit — one revolution per second. Bridges: 1 Hz = 2π rad/s = 60 rpm. 1 rpm = 2π/60 rad/s ≈ 0.1047 rad/s. The 2π factor is the radian-vs-revolution conversion (2π radians = one full turn).
1 Hz = 2π rad/s = 60 rpm
Where each is used.
rad/s in physics, control theory, signal processing. rpm in automotive (engine idle ~700 rpm, redline ~7000 rpm), HVAC fans, CD/ DVD players, industrial motors. Hz in electrical engineering (mains frequency 50 / 60 Hz), AC machinery, processor clocks (rotational frequency of clock domain). Same physical quantity; convention picks which notation by industry.
Tangential speed.
Angular velocity times radius gives the linear speed at the edge: v = ω r. A turntable spinning at 33⅓ rpm (LP record): ω = 33.33 × 2π/60 = 3.49 rad/s. Outer edge of a 30 cm record: v = 3.49 × 0.15 ≈ 0.52 m/s. Inner edge of the music groove (about 6 cm radius): v ≈ 0.21 m/s. The radius difference is why a long-playing vinyl sounds slightly worse at the inner grooves — the stylus is reading slower-moving signal per second.
Common rotational speeds.
Earth's rotation: 7.27 × 10⁻⁵ rad/s (1 turn in 24 hours). Helicopter rotor: 250-500 rpm. Car engine: 700-7000 rpm. Hard disk drive: 5400 / 7200 / 10,000 rpm; the rate determines seek time. Centrifuge: 10,000-100,000 rpm. Turbocharger turbine: 100,000-300,000 rpm. Dental drill: 400,000 rpm. The range spans 10 orders of magnitude; the unit you'd pick depends on which end you're at.
Tachometer reading.
The dial in a car shows RPM in thousands ("x1000 rpm"). The redline is the safe limit, set by valvetrain dynamics and bearing tolerance. Modern engines rev to 6500-7500 rpm; Formula 1 engines pre-2014 hit 18,000+ rpm. Diesel engines redline lower (4000-5000 rpm) because larger bore and stroke mean higher piston speeds at the same RPM. The RPM number's significance depends entirely on the engine architecture; raw numbers across engine types aren't comparable.