SCIENCE · INVESTIGATE AND EXPLAIN

Read an angular encoder without losing a revolution or reversal

Angular position requires a reference and a positive rotation direction. This packet starts at an unwrapped350° with positive counterclockwise rotation. An encoder reports the same direction as an angle between 0° and 360°, but that wrapped coordinate discards the number of full turns. A jump from 350° to 20° can be a coordinate wrap rather than a 330° clockwise rotation.

Learning outcome: Compare a steady wrap, an interior reversal and a zero-velocity endpoint. Preserve raw wrapped samples, disclosed unwrapped coordinates, signed displacement and swept angle. Explain which conclusions rely on the known generator rather than the samples alone.

Starting recap

Declare the axis, angular units, recipient and valid contact or material branch. Distinguish supplied shape factors from independently measured inertia.

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Try

With ω0=30°/s and α=0, the final unwrapped angle at 4 s is 470° but the encoder reads110°. The true displacement is 120°, whereas simply subtracting110−350 would give−240°.

Open the experiment

Predict

An encoder coordinate wraps from 350° to 20°. Without more information this alone:

Your prediction and reason

Experiment

Compare a steady wrap, an interior reversal and a zero-velocity endpoint. Preserve raw wrapped samples, disclosed unwrapped coordinates, signed displacement and swept angle. Explain which conclusions rely on the known generator rather than the samples alone.

Compare wrapped readouts with unwrapped angular motion

Worked teaching model · synthetic values · Fixed-axis point starts at unwrapped angle 350 degrees; constant selected alpha holds over 0–4 s. Instrument reports angles modulo 360 degrees at integer seconds. Generator is disclosed; wrapped samples alone do not uniquely establish accumulated revolutions.

Initial angular velocity: 60 degrees/s Constant angular acceleration: -30 degrees/s²

Reset keeps your prediction, captures and practice history.

Disclosed unwrapped model and sampled encoder readings

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: unwrapped generator (solid).2: wrapped encoder samples (filled markers). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Disclosed unwrapped model and sampled encoder readings

unwrapped generator

time (s)	angle (degrees)
0	350
0.05	352.962
0.1	355.85
0.15	358.663
0.2	361.4
0.25	364.063
0.3	366.65
0.35	369.163
0.4	371.6
0.45	373.962
0.5	376.25
0.55	378.462
0.6	380.6
0.65	382.663
0.7	384.65
0.75	386.563
0.8	388.4
0.85	390.163
0.9	391.85
0.95	393.462
1	395
1.05	396.462
1.1	397.85
1.15	399.163
1.2	400.4
1.25	401.563
1.3	402.65
1.35	403.663
1.4	404.6
1.45	405.462
1.5	406.25
1.55	406.962
1.6	407.6
1.65	408.163
1.7	408.65
1.75	409.063
1.8	409.4
1.85	409.663
1.9	409.85
1.95	409.962
2	410
2.05	409.963
2.1	409.85
2.15	409.663
2.2	409.4
2.25	409.063
2.3	408.65
2.35	408.163
2.4	407.6
2.45	406.962
2.5	406.25
2.55	405.462
2.6	404.6
2.65	403.663
2.7	402.65
2.75	401.563
2.8	400.4
2.85	399.163
2.9	397.85
2.95	396.462
3	395
3.05	393.462
3.1	391.85
3.15	390.163
3.2	388.4
3.25	386.563
3.3	384.65
3.35	382.663
3.4	380.6
3.45	378.462
3.5	376.25
3.55	373.962
3.6	371.6
3.65	369.163
3.7	366.65
3.75	364.063
3.8	361.4
3.85	358.663
3.9	355.85
3.95	352.962
4	350

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

wrapped encoder samples

time (s)	angle (degrees)
0	350
1	35
2	50
3	35
4	350

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Final unwrapped angle	350	degrees
Final wrapped encoder readout	350	degrees
Signed angular displacement	0	rad
Total swept angle	2.0944	rad
Final angular velocity	-1.0472	rad/s
Average angular velocity	0	rad/s
Interior reversal exists	1	flag
Interior reversal time	2	s

A wrap in the displayed coordinate is not necessarily a physical reversal.

For constant alpha, theta=theta0+omega0 t+alpha t²/2 and omega=omega0+alpha t. Convert degrees by pi/180 before using arc-length or SI dynamic formulas. Split swept angle at an interior zero of omega; a zero exactly at the endpoint adds no reversed segment. These known generator settings resolve the history, but unqualified sparse modulo readings alone need not do so.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

A wrap in the displayed coordinate is not necessarily a physical reversal.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Choose the axis, reference or material model

Build the algebraic relationship

For constant angular acceleration, use θ=θ0+ω0t+αt²/2 and ω=ω0+αt. These equations parallel constant-acceleration straight-line motion, but the coordinate is angular. In SI calculations convert degrees to radians by multiplying byπ/180. One revolution is 2π radians; using a degree value directly in arc length rθ would be incorrect.

Check the original numerical packet

The disclosed generator holds the selected acceleration for 4 s. With ω0=60°/s and α=−30°/s², the angular velocity reaches zero at 2 s. Unwrapped angle rises from 350° to 410° and returns to 350°. Signed displacement and average angular velocity are zero, but total swept angle is 120°=2π/3 rad. The final angular velocity is−60°/s, not zero.

Test the boundary and independent evidence

Total swept angle adds the forward and backward portions after locating the reversal. A zero velocity exactly at the observation endpoint adds no reversed segment. The supplied generator fixes this history, but sparse wrapped samples alone need not identify unique accumulated rotations. An independently justified speed bound or additional observations would be needed to unwrap an unknown record reliably.

Worked original case

With ω0=30°/s and α=0, the final unwrapped angle at 4 s is 470° but the encoder reads110°. The true displacement is 120°, whereas simply subtracting110−350 would give−240°.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

A fixed-axis body starts at angular velocity2 rad/s with constant angular acceleration−1 rad/s² for 3 s. Find its total swept angle in radians.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All records are supplied or generated teaching evidence. No rotating equipment, applied loads, material tests or physical assembly is requested. An admissible virtual model state does not certify real equipment safety, supervised practical skill or independent mastery.

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Public practice checks a specific claim. It does not establish independent mastery.

Why is a change of angle convention not a physical change of rotational direction?

Your explanation and limitations

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