Read rotational graphs: slope and signed area
You will be able to: Connect θ–t, ω–t and α–t graphs through slopes and signed areas.
How do rotational graphs describe a turn and a reversal?
A wheel’s ω–t graph rises from 2 to 6 rad/s in 2 s, then falls to −2 rad/s over the next 2 s. The wheel first speeds up, then slows and eventually turns the other way. The graph tells us both when that happens and the net angle turned.
A useful starting point: Angular velocity and acceleration: keep the signs →
Words and symbols before equations
- Slope
- Vertical change divided by horizontal change.
- Signed area
- Area above the horizontal axis adds; area below subtracts.
- Angular-position graph
- θ in rad versus time in s; its local slope is ω.
- Angular-velocity graph
- ω in rad/s versus time in s; its local slope is α.
What this picture assumes
ω rises linearly from 2 to 6 rad/s during 0–2 s, then falls linearly to −2 rad/s during 2–4 s. The plot shows only the interval up to the selected time. The slope changes abruptly at 2 s.
Connect the picture to the physics
The slope of θ–t gives angular velocity. A flat θ–t segment means no change in angle during that segment. For a curve, local steepness matters, not just the average slope between distant points.
The slope of ω–t gives α. Its signed area gives Δθ. For a straight ω–t segment, use a trapezoid: average of endpoint angular velocities times duration. Split at a zero crossing if calculating total angular travel.
The signed area under α–t gives Δω. At a sharp corner in an ideal ω–t graph, describe the slopes just before and after; one instantaneous slope is not defined at the corner. A zero crossing of ω signals a reversal if the sign changes.
A worked example, step by step
For the wheel described above, find the accelerations, reversal time and net angular displacement from 0 to 4 s.
- During 0–2 s, α=(6−2)/2=+2 rad/s². During 2–4 s, α=(−2−6)/2=−4 rad/s².
- After t=2 s, ω=6−4(t−2). Set ω=0: t=3.5 s.
- First area: [(2+6)/2](2)=8 rad. Second signed area: [(6−2)/2](2)=4 rad.
- Net Δθ=12 rad. Total angular travel is 13 rad: the final negative-area triangle has magnitude 0.5 rad, which adds to distance rather than subtracting.
Area gives displacement, not automatically total angular travel when the rotation reverses.
Can an ω–t graph have positive height and negative slope?
Compare with an explanation
Yes. The wheel is rotating counterclockwise but slowing down at that instant.
Predict. Change one thing. Explain.
Move the time control through 2 s, 3.5 s and 4 s. Identify the slope change, momentary stop and reversal. Compare the signed accumulated area with angular displacement.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At t=3 s: ω=2 rad/s; accumulated Δθ=12 rad. Current α=-4 rad/s².
ω rises linearly from 2 to 6 rad/s during 0–2 s, then falls linearly to −2 rad/s during 2–4 s. The plot shows only the interval up to the selected time. The slope changes abruptly at 2 s.
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 questionAn ω–t line falls from +4 to −4 rad/s over 4 s. (a) Find α. (b) Find reversal time. (c) Find net Δθ. (d) Find total angular travel.
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Compare with the answer and four-point rubric
- 1 point: α=(−4−4)/4=−2 rad/s².
- 1 point: t=2 s.
- 1 point: Zero, because equal positive and negative triangle areas cancel.
- 1 point: Each triangle has area ½(2)(4)=4 rad, so total travel is 8 rad.
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 1Slope of ω–t?
Angular acceleration.
RECALL 2Signed area of ω–t?
Angular displacement.
RECALL 3How do you find total angular travel?
Add the magnitudes of the areas, splitting at reversals.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read rotational graphs: slope and signed area
- Slope of θ–t → ω; slope of ω–t → α.
- Signed area under ω–t → Δθ; under α–t → Δω.
Remember: Area gives displacement, not automatically total angular travel when the rotation reverses.
Conditions: ω rises linearly from 2 to 6 rad/s during 0–2 s, then falls linearly to −2 rad/s during 2–4 s. The plot shows only the interval up to the selected time. The slope changes abruptly at 2 s.
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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