Writing a reproducible physics research report

Use an ideal simple pendulum: a point bob on a massless inextensible support, a fixed frictionless pivot and uniform gravitational acceleration. The length runs from pivot to the bob’s center of mass. Let tau be the elapsed time for an explicitly counted integer number of complete cycles. The period is tau divided by that count. In the small-angle approximation, gravity is proportional to length and to the square of count divided by elapsed time.

A raw-data record should preserve those inputs rather than only a computed period. Count a cycle between equivalent phase crossings so that a half swing cannot be mistaken for a full oscillation. Record the reference for length and any timebase calibration. Under independent small input uncertainties, logarithmic differentiation gives the leading relative uncertainty of the gravity estimate. A wrong cycle count is a discrete metadata error, not a tiny continuous fluctuation to include in that formula.

The exact undamped pendulum period follows from energy conservation and a one-dimensional integral. Substituting sine of half the turning angle converts it into a complete elliptic integral. Expanding the integrand at small turning angle and averaging its sine-squared factor gives a period increase of angle squared over sixteen. This derives the leading finite-amplitude effect from the same ideal pendulum mechanics rather than assigning an arbitrary systematic allowance.

Because the measured finite-amplitude period is longer than the small-angle period, inserting it directly into the small-angle gravity formula underestimates gravity. At fixed measured period, the leading corrected estimate therefore increases by angle squared over eight in fractional terms. The sign matters. A corrected report should state the retained order and account for amplitude uncertainty if relevant. Air drag, a finite bob, support elasticity and pivot motion are other potential model changes with different observable signatures.

An auditable result states the estimated quantity, its units, uncertainty convention, applied corrections and validity range. Separate illustrative data from actual observations and name the primary source of the physical model. If code performs the reduction, preserve its version, constants, unit conversion and input data together with sufficient arithmetic that a short case can be checked by hand. A reproducible numerical answer need not be physically accurate if the measurement model or metadata is wrong.

The final claim should distinguish a derivation of the amplitude correction from an empirical gravity estimate and from a proposal to improve the apparatus. A comparison run at another length tests the predicted square-root period scaling and can reveal count or length errors. A small residual then has a concrete interpretation rather than serving as decoration. The worked example shows a correction larger than the quoted measurement uncertainty, while the practice example demonstrates a metadata mistake that no increase in numerical precision could repair.