Measure gravity with pendulum timing
You will be able to: Design a pendulum experiment and obtain g from a linearized graph.
How can several measurements give a better test than one stopwatch reading?
A student times ten swings for each of several pendulum lengths. Instead of calculating gravity from one trial, they plot period squared against length. A straight-line trend uses all the measurements and makes inconsistent points easier to notice.
A useful starting point: A small-angle pendulum: length and gravity set the timing →
Words and symbols before equations
- Linearize
- Transform variables so a model predicts a straight-line graph.
- Slope S
- Vertical change divided by horizontal change, with units.
- Repeat trials
- Measurements under the same conditions to estimate variability.
- Measurement uncertainty
- A range describing the limits of a measurement, not a guarantee of exact accuracy.
What this picture assumes
Exact calculated small-angle pendulum data at lengths 0.25, 0.50, 0.75 and 1.00 m. No measurement noise. These are synthetic model points, not real student measurements or evidence of experimental accuracy.
Connect the picture to the physics
Square T=2π√(ℓ/g) to obtain T²=(4π²/g)ℓ. Plot T² in s² vertically and length ℓ in m horizontally. The slope S has units s²/m and equals 4π²/g; therefore g=4π²/S. The ideal graph passes through the origin.
Measure length from pivot to bob center, use a small consistent release angle, and release without a push. Time N full cycles using the same reference point and direction, then compute T=Δt/N. Repeat at each length, report the scatter, and collect a spread of lengths while holding the bob and timing method fixed.
Fit a line to the trend rather than choosing two neighboring noisy points. An unexpected intercept can suggest a length offset or other systematic issue; do not force it to zero without examining the measurements. More repetitions reduce random variability but cannot repair measuring to the wrong part of the bob.
A worked example, step by step
A best-fit line of T² versus ℓ has slope 4.0 s²/m. Estimate g. Explain why using a raw T-versus-ℓ slope would not work the same way.
- Use T²=(4π²/g)ℓ, so S=4π²/g.
- Rearrange: g=4π²/S.
- g≈39.48/4.0=9.87 m/s².
- T versus ℓ is a square-root curve, so its slope is not the constant 4π²/g. The transformed axes matter.
A clean synthetic graph is not real experimental evidence. Real measurements need repeated trials and uncertainty assessment.
If the T²-versus-ℓ slope is larger, is the inferred g larger?
Compare with an explanation
No. g=4π²/S, so a larger slope corresponds to smaller g.
Predict. Change one thing. Explain.
Change g in the synthetic model and predict the line’s slope. Inspect the labeled sample values. These exact calculated points demonstrate the model; they are not collected student measurements.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Slope=3.948 s²/m; inferred g=10 m/s². Synthetic (ℓ, T²) pairs: (0.25 m, 0.987 s²); (0.5 m, 1.974 s²); (0.75 m, 2.961 s²); (1 m, 3.948 s²). The origin is a model extrapolation.
Exact calculated small-angle pendulum data at lengths 0.25, 0.50, 0.75 and 1.00 m. No measurement noise. These are synthetic model points, not real student measurements or evidence of experimental accuracy.
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 questionPlan a small-angle pendulum experiment. (a) Name the independent and measured variables. (b) Describe a timing method that reduces relative reaction-time uncertainty. (c) State graph axes and the slope relationship. (d) Identify one systematic error and a correction.
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Compare with the answer and four-point rubric
- 1 point: Vary pivot-to-bob-center length; measure elapsed time for a known number of full cycles and calculate T.
- 1 point: Time multiple full cycles from the same phase, repeat trials, and divide elapsed time by cycle count.
- 1 point: Vertical T² (s²), horizontal ℓ (m); slope=4π²/g, so g=4π²/slope.
- 1 point: Example: measuring only string length misses the bob-center offset; measure from pivot to the center of mass.
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 axes linearize the pendulum period law?
T² versus ℓ.
RECALL 2How do you infer g from slope S?
g=4π²/S.
RECALL 3Do repeated timings remove a systematic length error?
No; correct the measurement method.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Measure gravity with pendulum timing
- T²=(4π²/g)ℓ.
- For T² vs ℓ: slope S=4π²/g; g=4π²/S.
Remember: A clean synthetic graph is not real experimental evidence. Real measurements need repeated trials and uncertainty assessment.
Conditions: Exact calculated small-angle pendulum data at lengths 0.25, 0.50, 0.75 and 1.00 m. No measurement noise. These are synthetic model points, not real student measurements or evidence of experimental accuracy.
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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