How this instrument works
Uniform circular motion runs on one shared number. Every point of a rigid spinning body sweeps identical angle each second, so ω belongs to that whole body — while tangential speed and centripetal acceleration belong to individual points, both scaling with distance from an axis. Speed grows in simple proportion, v = ω·r. Inward acceleration grows with radius too, a_c = ω²·r, though spin rate enters squared. Bonded abrasive discs live by that first relation: EN 12413 caps peripheral speed near 80 m/s, which is why a 125 mm grinder disc is stamped 12,200 rpm while a 230 mm disc, swung on a larger machine, is stamped 6,600 rpm. Nearly double that radius, barely half that spin rate, and rim speed lands in identical territory.
That squared dependence was engineered before anyone taught it. Centrifugal governors — adapted by James Watt and Matthew Boulton in 1788 from gear already regulating flour mills — hang two heavy balls off a rotating spindle and let ω²r do arithmetic mechanically. Balls swing outward until their cone sits at whatever angle lets gravity balance that outward demand, and a linkage then lifts to throttle steam. Geometry there yields one startling result: vertical height of that cone equals g ⁄ ω², independent of arm length and independent of ball mass. Spin a governor at 5 rad/s and its cone stands 0.392 m tall, whatever hardware hangs from it. Watt held engine speed steady by measuring a length. Maxwell's 1868 paper 'On Governors', written to explain why some of them hunted rather than settling, founded control theory.
Both lines assume plenty. Radius must hold fixed, ruling out spirals and stretchy tethers that lengthen under load. Angular velocity must be steady: let it change and a tangential term α·r appears beside that radial one, so total acceleration stops aiming at any centre. Radius must be measured from an axis and square to it, never from some convenient bracket. And a_c is acceleration, never force — no mass appears in either line, so nothing here knows whether a 3-gram sample or a 3-tonne rotor is doing that turning. Multiply by mass when newtons are wanted. Reference frame matters as well: these figures describe what a stationary observer sees. Ride along instead and you feel outward push, identical physics kept in different books.
- Enter Radius measured from your rotation axis out to whichever point you are asking about — centimetres, metres, or kilometres.
- Enter Angular velocity for that spin. Its menu takes rad/s, deg/s, or rpm, so a motor nameplate figure goes in exactly as printed.
- Read Tangential speed in m/s, km/h, or mph. That is how fast your chosen point genuinely travels along its path.
- Read Centripetal acceleration in m/s², ft/s², or multiples of g0 — quickest way to judge whether a mounting or a bearing will cope.
- Audit by eye: Centripetal acceleration should equal Tangential speed times Angular velocity. Any other ratio means a unit menu is set wrong.
Worked example — a whirling arm at two metres
Whirling arms were aerodynamics before wind tunnels existed: Benjamin Robins swung one in 1746, and George Cayley used his in 1804 to measure lift on flat plates. Build a teaching version with a 2 m arm turning at 5 rad/s. Set Radius to 2 and Angular velocity to 5. Tangential speed returns v = 5 × 2 = 10 m/s. Centripetal acceleration returns a_c = 5² × 2 = 25 × 2 = 50 m/s². Both fall out exact by inspection, with no rounding anywhere.
So a model bolted at that arm tip meets 10 m/s of airflow, about 36 km/h, while its mounting carries 50 m/s² inward — 5.1 times standard gravity, which is why whirling-arm hardware always looks overbuilt for its size. Two cross-checks agree: v² ⁄ r = 100 ⁄ 2 = 50, and v·ω = 10 × 5 = 50. Five radians per second is 47.75 rpm, so one lap takes 2π ⁄ 5 = 1.257 s, and 12.57 m of circumference covered at 10 m/s matches that exactly.
Wind tunnels displaced these rigs for one reason no calculator can show. After a single revolution your model flies back into its own wake, so airflow past it is no longer a clean 10 m/s — a systematic error that dogged rotary-arm lift data until Francis Wenham built his tunnel in 1871.
Questions
Why is a 230 mm grinding disc rated for fewer rpm than a 125 mm one?
Because ratings cap rim speed, not spin rate. Bonded abrasives burst once v = ω·r passes roughly 80 m/s, a limit written into EN 12413, so permitted ω falls as radius rises: 0.0625 m of radius allows about 1,280 rad/s (12,200 rpm), while 0.115 m allows about 696 rad/s (6,600 rpm). Fit an oversized disc to a fast machine and rated rim speed is exceeded long before anything sounds wrong. Worn discs grow safer for exactly that same reason — shrinking radius drops rim speed at unchanged rpm.
Why does mass appear nowhere in either formula?
Because both outputs are kinematic: they describe motion, not whatever causes it. A dust grain and a turbine wheel sitting at identical radius on identical spin share identical tangential speed and identical centripetal acceleration. Mass enters only when force is wanted, through F = m·a_c = m·ω²r, which is what a centripetal force instrument computes. Keep those roles apart — this sheet reports how violently something turns, and mass reports what that costs in newtons.
What changes if I double Angular velocity instead of Radius?
Doubling Radius doubles both outputs, since v and a_c are each strictly proportional to r. Doubling Angular velocity doubles Tangential speed but quadruples Centripetal acceleration, because ω enters that second line squared. Practical upshot: moving a sensor twice as far out is a mild change, whereas spinning twice as fast is a violent one. Rotor designers feel this daily — a 20% overspeed adds 20% to rim speed and 44% to radial loading.
Where exactly should Radius be measured from?
From your rotation axis, perpendicular to it, out to whichever point is under question. Two slips are common. First, entering diameter: a 300 mm wheel has Radius 0.15 m, not 0.3. Second, measuring along a slanted line instead of square to that axis — on a coning helicopter blade or a tilted governor arm, only perpendicular distance counts. Points nearer an axis genuinely do travel slower, so choose your point deliberately rather than defaulting to a rim.
Does this still hold while rotation is speeding up?
Tangential speed stays right for that instant, but Centripetal acceleration becomes only part of a larger vector. Changing ω adds a tangential term α·r, where α is angular acceleration in rad/s², and total acceleration grows to √((ω²r)² + (αr)²), aimed somewhere between inward and forward. During spin-up that extra term can dominate. Once ω settles, it vanishes and these two lines are complete again, so enter whichever ω holds at whatever moment interests you.
Is Centripetal acceleration what an accelerometer on that arm would read?
Close, with one sign convention worth pinning down. Centripetal acceleration points inward, toward your axis, and 50 m/s² means precisely that. An accelerometer riding along measures proper acceleration, so it reports outward specific force holding it in place — same 50 m/s², or 5.1 g, with an outward arrow. Identical magnitude, opposite direction, each correct within its own frame. Divide by 9.80665 for g, or simply switch that output menu to g0.