A small-angle pendulum: length and gravity set the timing
You will be able to: Apply the small-angle pendulum model and state its limitations.
Why does a longer playground swing have a longer period?
A long swing takes more time to go out and back than a short one at the same location. In the ideal small-angle model, the bob’s mass does not set that timing. Length and gravitational acceleration do.
A useful starting point: What sets a spring oscillator’s period? →
Words and symbols before equations
- Simple pendulum
- A point-like bob on a light, fixed-length string with negligible resistance.
- Length ℓ
- Distance from pivot to the bob’s center of mass, in m.
- Angle θ
- Angular displacement from vertical, measured in radians in equations.
- Small-angle approximation
- For sufficiently small angles, sinθ≈θ; this makes the restoring torque approximately proportional to θ.
What this picture assumes
Simple pendulum with a point-like bob and light inextensible string. Negligible resistance; small-angle period approximation. Drawing shows a fixed 10° snapshot, with length scale 40 drawing units/m. Changing controls compares separate trials.
Connect the picture to the physics
Gravity supplies a restoring torque τ=−mgℓ sinθ about the pivot; tension passes through the pivot and contributes no torque there. With small θ, τ≈−mgℓθ. Using I=mℓ² gives angular acceleration α≈−(g/ℓ)θ. The mass cancels.
The resulting small-angle period is T=2π√(ℓ/g). Quadrupling length doubles T; stronger gravity shortens T. The formula is approximately independent of amplitude only while angles remain small. Large swings are periodic but are not accurately described by this simple harmonic approximation.
At the bottom, tangential restoring acceleration is zero but a moving bob still has inward radial acceleration v²/ℓ. Its total acceleration is therefore not zero there. The one-dimensional oscillation statements concern the tangential motion; do not discard the radial force needed for a curved path.
A worked example, step by step
Use g=10 m/s². Compare small-angle pendulums with lengths 1 m and 4 m.
- T_1=2π√(1/10)≈1.99 s.
- T_2=2π√(4/10)≈3.97 s.
- Length quadrupled, so period doubled.
- Changing bob mass alone would not change either ideal period. Measure length to the bob’s center.
A pendulum at the bottom can have nonzero radial acceleration even though its tangential restoring acceleration is zero.
Would doubling bob mass double the ideal small-angle period?
Compare with an explanation
No. Gravitational restoring torque and rotational inertia both scale with mass, so the mass cancels.
Predict. Change one thing. Explain.
Compare lengths at fixed g, then change g at fixed length. The drawing uses a fixed small-angle snapshot and labels the physical length; it does not exaggerate the swing angle. Predict the period factor each time.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
ℓ=1 m, g=10 m/s². Small-angle T≈1.987 s and f≈0.503 Hz. Bob mass does not enter this ideal timing relation.
Simple pendulum with a point-like bob and light inextensible string. Negligible resistance; small-angle period approximation. Drawing shows a fixed 10° snapshot, with length scale 40 drawing units/m. Changing controls compares separate trials.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy 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 small-angle pendulum has ℓ=0.25 m and g=10 m/s². (a) Find T. (b) Predict T at ℓ=1 m. (c) State the effect of doubling bob mass. (d) Explain why the formula may fail for a very large release angle.
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Compare with the answer and four-point rubric
- 1 point: T=2π√(0.025)≈0.993 s.
- 1 point: Approximately 1.99 s: length quadruples and period doubles.
- 1 point: No change in the ideal period.
- 1 point: sinθ is no longer approximately θ, so the restoring torque is not proportional to angle over the motion.
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 1Which length enters the pendulum formula?
Pivot to bob’s center of mass.
RECALL 2Which two quantities set the small-angle period?
Length and gravitational acceleration.
RECALL 3Why is the small-angle condition needed?
It makes sinθ≈θ, yielding an approximately linear restoring torque.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A small-angle pendulum: length and gravity set the timing
- Small-angle simple pendulum: T=2π√(ℓ/g).
- τ≈−mgℓθ for small θ in radians.
- At the bottom: a_t=0 but a_radial=v²/ℓ if moving.
Remember: A pendulum at the bottom can have nonzero radial acceleration even though its tangential restoring acceleration is zero.
Conditions: Simple pendulum with a point-like bob and light inextensible string. Negligible resistance; small-angle period approximation. Drawing shows a fixed 10° snapshot, with length scale 40 drawing units/m. Changing controls compares separate trials.
Refresh Kid · Unit 7 · Objectives 7.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 7.2, objectives 7.2.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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