Angular position and displacement

An angle is a measurement against something, so name that something first. Choose the rotation axis, a reference line perpendicular to it and fixed in your frame, and a line fixed in the body, such as a spoke or a paint mark. The angular position θ is the angle from the reference line to the body line. Then choose the positive sense. The usual convention is counterclockwise as seen by an observer looking back along the axis from its positive end, which is the right-hand rule that later turns rotation into a vector. The sign belongs to the viewpoint, not to the motion: a clock's minute hand sweeping from 12 to 3 turns through −π/2 rad seen from the front of the dial, and +π/2 rad seen from behind it. Neither is wrong, but an answer that does not say which side it was viewed from is incomplete.

For a given turn, the arc s that a point sweeps grows in proportion to its distance r from the axis, so s/r depends on the turn alone. That ratio is the angle in radians: θ = s/r. A full turn sweeps the circumference, so it is 2πr/r = 2π rad, and 1 rad = 360°/2π = 57.30°. Because θ is a length over a length it carries no dimension; rad is a label, not a unit that has to cancel. Only in radians is s = rθ true without a conversion factor, and only in radians does sin θ ≈ θ for small angles and d(sin θ)/dθ = cos θ, which is why calculus-based rotation is written in radians throughout. A point 0.34 m from a bicycle axle turning through 90° sweeps s = 0.34 m × π/2 = 0.53 m. Put θ = 90 into s = rθ instead and it returns 31 m, a result that fails the plausibility check at a glance.

Angular displacement is the change in angular position, Δθ = θ₂ − θ₁. Moving the reference line changes every θ and no Δθ. Measured from the wall, a door opens from θ₁ = 0.20 rad to θ₂ = 1.45 rad, so Δθ = 1.25 rad. Measure from a line 0.60 rad further round instead and the readings become −0.40 rad and 0.85 rad; the displacement is still 1.25 rad. This is the rotational twin of choosing an origin for x, and it is why the rates built later from differences in θ never depend on where θ = 0 was put. The sign of Δθ does depend on the positive sense, and it records the direction of the net rotation. Like its linear counterpart it is net: a rotor that turns +3.0 rev and then −1.0 rev has Δθ = +2.0 rev = 12.6 rad, although it swept 4.0 rev, or 25.1 rad, of angle along the way.

Add a whole turn to θ and the body looks exactly the same, so θ and θ + 2πn name one orientation for every integer n. When only the orientation matters, reduce θ into a single turn with θ mod 2π = θ − 2π⌊θ/(2π)⌋, which lands in [0, 2π). An angle of 23.0 rad contains ⌊23.0/6.283⌋ = 3 whole turns, so it reduces to 23.0 − 6π = 4.15 rad. An angle of −4.00 rad has ⌊−0.637⌋ = −1, so it reduces to −4.00 + 2π = 2.28 rad. Some problems prefer the range (−π, π], the smallest rotation from the reference line either way, where 4.15 rad becomes 4.15 − 2π = −2.13 rad. Reduction is a statement about how the body looks. It says nothing about how far the body has turned, and applied to a displacement it discards exactly the whole turns you were asked for.

Displacement is where periodicity must be refused. A winch drum of radius 0.15 m that turns through 5.40 rev pays out s = rΔθ = 0.15 m × 5.40 × 2π = 5.09 m of cable. Its final orientation, 5.40 rev reduced to 0.40 rev, would claim only 0.38 m. So θ is treated as a continuous function of time that climbs past 2π and 4π, or runs negative, exactly as x does on a long straight track. The number of revolutions is N = Δθ/2π, and it need not be a whole number. Continuity is also what will make dθ/dt meaningful in the next topic: a θ that dropped from 2π back to 0 at every lap would imply an infinite rate of turning at an instant when nothing at all happened to the drum.

Many sensors report only the reduced angle. An absolute encoder reads φ in [0, 2π), so its log jumps from about 6.2 rad to about 0.1 rad every lap, and the continuous θ has to be rebuilt. Take each change between samples, Δφ, and remove whole turns until |Δθ| ≤ π: Δθ = Δφ − 2πn with n the integer nearest Δφ/(2π). A step from 5.90 rad to 0.40 rad has Δφ = −5.50 rad, and −5.50/2π = −0.88 rounds to −1, so Δθ = −5.50 + 2π = +0.78 rad. The rule rests on an assumption the readings cannot test: that the body turned less than half a turn between samples. A true step of +7.07 rad, or of −5.50 rad, would leave the same two readings. Sampling fast enough that no half turn fits between readings is what makes the reconstruction unique, and the film wagon wheel that seems to spin backwards is that assumption failing.