Measure pendulum periods and test a prediction

Ask a question the apparatus can answer

Does doubling pendulum length multiply its period by two, or by approximately √2? Test 0.50 m and 1.00 m lengths while keeping the bob, pivot, release angle and timing method as consistent as possible. Write your prediction before collecting timings. The small-angle model predicts a ratio of √2 ≈ 1.414; it does not predict a factor of two.

This lesson offers a physical procedure and a synthetic practice alternative. The browser controls calculate model predictions; they do not measure a pendulum or calculate statistics from text in your notebook. Keep your raw readings on paper or in a local document, then summarize them in capture notes. Do not relabel the practice numbers as observations you collected.

Build and check the setup

Use a securely attached small, soft bob, light string, a stable support, a ruler or tape, a protractor or angle guide, and a stopwatch. Keep the swing path clear and low, away from faces and breakable objects. Check the support and attachment before release. Use a classroom setup approved by your instructor when available; do not improvise a heavy overhead weight.

Measure from the pivot to the center of the bob, not just to the knot. Record the length in metres, the smallest ruler division and your estimate of uncertainty in locating both endpoints. A 1 mm ruler division alone does not guarantee 1 mm length accuracy. Record the bob dimensions, support type and any visible stretching or pivot motion.

Set an initial angle near 10° from downward vertical. Release without pushing and keep the motion approximately in one plane. Note the angle-setting uncertainty. Do not adjust length while the pendulum is moving. If your apparatus cannot safely provide both lengths, use shorter lengths and calculate their predicted ratio separately; the browser only offers quarter-metre increments, so do not silently round an actual length to match a control.

Define the clock events before collecting data

Choose the lowest-point crossing in one direction as your reference. Let the bob complete an initial swing, then start timing as it crosses that reference in the selected direction. Call this crossing zero. Stop at the tenth later crossing in the same direction. The opposite-direction crossing is halfway through a cycle and must not be counted as a full one.

1. Measure ten complete cycles and retain the total time, including the displayed precision.
2. Stop the pendulum and reset the release angle. Make five separate timing trials for that length.
3. Repeat for the second length with the same bob and timing rule. Alternate which length is measured first across repeat sessions if practical, and record the order.
4. Record problems at the time they occur: a miscount, a pushed release, a moving support or a strongly elliptical path. Keep the original record and label any rejected trial with its reason before retaking it. An unexpected result alone is not a reason to discard a trial.

Five trials on one apparatus are repeated measurements of that apparatus, not five independent pendulum designs. A partner can count while you time, or you can exchange roles to explore observer effects. A video can help identify crossing events, but frame timing, perspective and event selection still introduce uncertainty. No upload is required.

Preserve raw evidence before averaging

For each trial record length, length uncertainty, release angle, cycle count, total time, timer display resolution, observer and any anomaly. Also retain the date, setup sketch and counting rule. Compute each period as T = t/N, where t is elapsed time in seconds and N is the number of full cycles. Average the five period estimates only after checking that all trials use the same counting rule.

For the first row, the total times sum to 71.0 s. Divide by five trials to obtain 14.2 s, then by ten cycles to obtain 1.420 s. For the second row, 100.5/5/10 = 2.010 s. The extra decimal in these calculated means does not establish millisecond accuracy. The period ratio is 2.010/1.420 ≈ 1.415, close to √2. These numbers were chosen for teaching; their agreement is not an experimental validation.

Keep uncertainty sources separate

If the whole ten-cycle interval has an estimated uncertainty of ±0.2 s, its contribution to a period estimate is ±0.2/10 = ±0.02 s. This is an illustrative uncertainty assumption, not a measured property of your stopwatch or your reaction time. If you instead estimate separate start and stop errors, explain how you combine them; do not silently treat one endpoint estimate as the uncertainty of the entire interval.

Timing more cycles reduces the contribution of a roughly fixed start/stop timing error per period. It does not correct a wrong length, a wrong cycle count, a tilted pivot or an incorrect model. Very long runs can also change amplitude through damping. Repeated trials describe scatter; their minimum and maximum are not guaranteed bounds on the true period or a confidence interval. A stopwatch’s display resolution and human event-timing error are different quantities.

For small relative changes, T ∝ √L means that a 2% length uncertainty contributes roughly 1% to the predicted period. This estimate assumes the small-angle model with gravity held fixed. A release-angle uncertainty changes the nonlinear prediction too. State which uncertainty sources you considered before interpreting a difference between prediction and observation.

Compare observation with a prediction

Set the model to 0.50 m and 10°, then to 1.00 m and 10°. The nonlinear reference periods are approximately 1.421209 s and 2.009893 s. Capture both settings. The model assumes g = 9.81 m/s², a point bob and no drag; it is not an exact description of your apparatus.

In each capture note, use a compact evidence record such as: “Source: own measurements [or synthetic practice]. L: … m; estimated length uncertainty: … m; angle: …; N: …; raw total times: … s; mean T: … s; timing uncertainty estimate and basis: …; model T: … s; discrepancy: … s; anomaly: ….” Capture notes are limited to 1,200 characters; retain fuller raw records separately. The note is learner-authored text, not an automatically validated measurement dataset.

Calculate residual = mean observed period − predicted period. For the synthetic 1.00 m example, 2.010 − 2.009893 ≈ +0.000107 s. That small residual does not justify seven-digit physical accuracy: the stated timing uncertainty and unmodeled effects are much larger. Agreement means “consistent at the precision examined,” not “proved exactly correct.” If your data disagree, examine counting, length endpoints, release method and model assumptions before changing or deleting readings.

Predict an unfamiliar length before testing it

Use your 0.50 m result to predict the period at 0.75 m with the same small angle: multiply by √(0.75/0.50) = √1.5. With the synthetic mean, the prediction is approximately 1.739 s. Write it down, then collect a new set of trials if you have apparatus. Do not revise the original prediction after seeing the new timings; record a revised explanation separately.

Check the timing transfer

A learner times 20 full cycles in 34.8 s and estimates ±0.2 s uncertainty for the whole interval. The period estimate is 34.8/20 = 1.74 s, with a timing contribution of ±0.01 s. It is compatible with the 1.739 s teaching prediction at that timing scale, but this alone does not account for length, angle or physical-model uncertainty.

Finish with a short claim, the raw-data source, your calculation, a competing explanation for any discrepancy and the next measurement you would improve. A physical-investigation record should also show the apparatus and actual observation procedure. Completing the browser notebook or using synthetic practice does not demonstrate physical measurement skill. A teacher or peer can review your procedure and reasoning; the page’s presence checks cannot certify them.

The OpenStax pendulum discussion supports the ideal timing model. The University of Massachusetts pendulum procedure describes measuring multiple cycles and repeated trials. The protocol and synthetic practice records here are original teaching material.