How this instrument works
Angular displacement answers, for rotation, exactly what distance = speed × time answers for a straight line. Hold angular velocity steady and angle accumulates in direct proportion to seconds elapsed: θ = ω·t. Radians keep that arithmetic honest, because a radian is arc length divided by radius — so multiplying θ by any radius hands straight back distance travelled along an arc. One radian spans 57.2958°, and a complete turn is 2π ≈ 6.2832 rad, which is why answers here so seldom land on tidy fractions of a revolution.
This quantity hides a subtlety no linear counterpart carries: finite rotations refuse to add like vectors. Yaw a book 90°, then roll it 90°; reverse that order and your book finishes facing somewhere else entirely. Euler proved in 1775 that any sequence of turns about a fixed point collapses into one rotation about some axis, and Olinde Rodrigues published explicit formulae for composing two of them in 1840 — three years ahead of Hamilton's quaternions, which dress up identical algebra. Only infinitesimal angular displacements behave as proper vectors, which is precisely why angular velocity, being a limit of those infinitesimals, does behave as one.
Two conditions bind this sheet: constant rate, and one fixed axis. Where spin builds or fades you want θ = ω₀t + ½αt² instead, though feeding in an average ω still yields correct total sweep — just nothing about what happened partway through. Let an axis wander and no scalar can say where a body ended up. Third caution is far and away most common in workshops: output here is accumulated angle, never final orientation. Ten radians is 572.96°, which is a whole revolution plus 212.96°. Protractors, compass roses and single-turn encoders all report that leftover 212.96° and silently discard a revolution. Anything that cares about cable twist, thread engagement or winding layers needs an accumulated figure, which is why multi-turn absolute encoders exist.
- Enter Angular velocity — how fast rotation proceeds. Its unit menu takes rad/s, deg/s or rpm, so a tachometer or nameplate figure goes in as printed.
- Enter Time elapsed, choosing milliseconds, seconds, minutes or hours. Negative values are refused; a clock does not run backwards.
- Read Angle swept to six decimals, then switch its unit among radians, degrees and turns. Turns reads most clearly when you want revolutions.
- Cross-check physically: Angle swept in radians times any radius in metres gives arc length in metres for a point at that distance from an axis.
Worked example — a five-second pan on a mast camera
A pan-tilt camera high on a harbour mast is driven from its joystick at a steady 2 rad/s — 114.59 deg/s, or 19.1 rpm, brisk yet still watchable on a monitor. An operator holds that joystick over for 5 seconds. Put 2 into Angular velocity and 5 into Time elapsed, and Angle swept returns θ = 2 × 5 = 10 radians, exact by inspection.
Ten radians is 1.5915 turns, or 572.96°. On-screen heading, though, would read 212.96°, because a whole revolution has already been folded away. That gap is what any pan controller must track: against a mechanical stop at 370° of travel, this camera reached its limit 3.23 seconds in and spent its last second-and-a-half going nowhere. Slip rings buy genuinely continuous panning by removing that stop altogether. Meanwhile a lens mounted 120 mm off its pan axis has traced s = rθ = 0.12 × 10 = 1.2 m of arc.
Questions
Is angular displacement the same as final orientation?
No — and confusing them is the standard slip. Angular displacement counts every radian swept, revolutions included; orientation is merely where a body points at one instant, folded into a single turn. Spin something through 10 rad and displacement is 10 rad, while a protractor laid against it reads 212.96°. Both are correct answers to different questions. Anything involving twist, wind-up, thread count or cable management needs accumulated angle.
How do I turn radians swept into revolutions?
Divide by 2π ≈ 6.28319. So 10 rad is 1.5915 turns, 100 rad is 15.915 turns, and π rad is exactly half a turn. Going to degrees, multiply radians by 180⁄π ≈ 57.2958. Angle swept carries all three units in its menu, so you can read radians for further calculation and turns for a mental picture without retyping anything.
What if angular velocity is not constant?
Then θ = ω·t needs replacing by θ = ω₀t + ½αt² for steady angular acceleration α, or by an integral of ω over time for anything messier. There is a useful escape, though: where you already know an average angular velocity across your interval, θ = ω·t still returns correct total sweep, since averaging is exactly what an integral divided by time means. What it cannot tell you is anything about the middle of that interval.
Can angular displacement be negative or larger than one turn?
Both, routinely. Sign follows a right-hand rule: curl fingers along rotation and a positive result points along your thumb, so an anticlockwise sweep viewed from above is positive and a clockwise one negative. Enter a negative Angular velocity and Angle swept comes back negative, meaning identical magnitude turned oppositely. Magnitude itself has no ceiling — a drill held 90 seconds at 250 rad/s racks up 22,500 rad, roughly 3,581 revolutions.
Why do finite angular displacements not add like vectors?
Because turns about different axes do not commute — order changes outcomes. Pitch a phone 90° nose-down, then yaw it 90°, and it finishes in a different attitude than yawing first and pitching second. Vector addition is commutative by definition, so finite turns cannot be vectors. Infinitesimal ones can, since order effects vanish to second order, and that is what rescues angular velocity. For real attitude bookkeeping, engineers reach for rotation matrices or quaternions rather than three stacked angles.
Why does my rotary encoder forget whole turns after a power cycle?
Because most absolute encoders are single-turn: they resolve position beautifully inside one revolution and hold no memory of how many revolutions came before. Incremental encoders fare worse, counting pulses from wherever they woke up, which is why machines run a homing routine at startup. Multi-turn absolute encoders fix it with gear trains or battery-backed counters, tracking accumulated displacement across thousands of revolutions — precisely what this sheet computes.