Measure a turn in radians
You will be able to: Relate a signed angular displacement to a turn and an arc length.
What does one radian look like?
Put a marker on the rim of a wheel. If it travels an arc whose length equals the wheel’s radius, the wheel has turned through one radian. The size of the wheel does not change what one radian means.
A useful starting point: Position and displacement →
Words and symbols before equations
- Axis of rotation
- The line about which the wheel turns; viewed here end-on.
- Angular displacement Δθ
- The signed angle turned, measured in radians (rad).
- Radian
- An angle defined by arc length divided by radius: θ=s/r.
- Rigid system
- A system that holds its shape as it turns.
What this picture assumes
Fixed viewing side, 0.4 m radius. The slider selects one signed turn from the starting ray, not elapsed time. The highlighted arc is traveled distance, not straight-line displacement.
Connect the picture to the physics
Choose a viewing side and a positive sense. Here counterclockwise (CCW) is positive and clockwise (CW) is negative. Reversing the viewing side reverses which apparent direction is clockwise, so keep the view fixed.
A full revolution has arc length 2πr. Dividing by r gives 2π radians, equivalent to 360°. Half a turn is π rad and a quarter turn is π/2 rad. Convert degrees by multiplying by π/180.
Use unwrapped angles when tracking multiple turns: one complete positive turn is a displacement of +2π rad even though the marker returns to its starting location. Net angular displacement includes signs; total angle traveled adds magnitudes for successive segments. For a single non-reversing turn, arc distance is r|Δθ|.
A worked example, step by step
A marker at radius 0.4 m makes a quarter turn CCW, then an eighth turn CW. Find net angular displacement and total arc distance traveled.
- The first turn is +π/2 rad; the second is −π/4 rad.
- Net Δθ=π/2−π/4=+π/4≈0.785 rad.
- Total angular travel is π/2+π/4=3π/4 rad.
- Arc distance=0.4(3π/4)=0.3π≈0.942 m. Net angle and total distance answer different questions.
Returning to the same marker position does not erase completed revolutions. Do not use degree values directly in radian-based formulas.
A wheel completes one full CCW revolution. Is Δθ zero?
Compare with an explanation
No. Its unwrapped angular displacement is +2π rad, although the marker ends in the same place.
Predict. Change one thing. Explain.
Change the signed angle from −180° to +180°. Compare degrees, radians and arc distance for a marker 0.4 m from the axis. The slider chooses a turn from zero; it is not an elapsed-time animation.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Signed turn=90° = 1.57 rad. At r=0.4 m, arc distance=0.63 m. Counterclockwise from the starting ray.
Fixed viewing side, 0.4 m radius. The slider selects one signed turn from the starting ray, not elapsed time. The highlighted arc is traveled distance, not straight-line displacement.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use an angular-motion or torque relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA wheel turns 180° CCW and then 90° CW, viewed from the same side. A marker is 0.2 m from the axis. (a) Convert each turn to signed radians. (b) Find net Δθ. (c) Find total angular travel. (d) Find arc distance traveled.
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Compare with the answer and four-point rubric
- 1 point: +π rad and −π/2 rad.
- 1 point: +π/2 rad.
- 1 point: 3π/2 rad.
- 1 point: 0.2(3π/2)=0.3π≈0.942 m.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1How many radians in one revolution?
2π.
RECALL 2What makes an angular displacement negative here?
Clockwise rotation from the specified viewing side.
RECALL 3Arc distance formula for one turn without reversal?
s=r|Δθ|, with θ in radians.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Measure a turn in radians
- 2π rad=360°; θ_rad=θ_deg π/180.
- For a non-reversing turn, arc distance=r|Δθ|.
Remember: Returning to the same marker position does not erase completed revolutions. Do not use degree values directly in radian-based formulas.
Conditions: Fixed viewing side, 0.4 m radius. The slider selects one signed turn from the starting ray, not elapsed time. The highlighted arc is traveled distance, not straight-line displacement.
Refresh Kid · Unit 5 · Objectives 5.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.1, objectives 5.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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