designing a reproducible physical investigation

A reproducible investigation is one in which an independent researcher, given only the written protocol, obtains the same result within the stated uncertainty. Design begins with a research question tied to physical theory: theory indicates which quantities matter and which may be neglected. Variables must be classified as independent, dependent and controlled, and every quantity needs an operational definition, that is, the measurement procedure that fixes its value. The measurement protocol states the instrument, its resolution, calibration, the number of trials $N$, and how the mean $\bar{x}$ and uncertainty $u$ are computed. Reproducibility requires recording raw data, environmental conditions and the analysis model so that others can repeat or refute the work. Finally, systematic error shifts all readings in one direction and is addressed by calibration and comparison of methods, while random error is reduced by averaging over trials, with uncertainty decreasing roughly as $u \propto 1/\sqrt{N}$.

Five independent length readings are given: 12.4, 12.5, 12.3, 12.5, 12.3 cm. Compute the mean value and the standard uncertainty of the mean.

Design a reproducible investigation of whether the period of a simple pendulum depends on amplitude at fixed length and mass. Identify the variables, the measurement protocol and the uncertainty estimate.

A report says: We measured the resistance with a multimeter and got about 10 ohms; the result matched theory. Rewrite this as a reproducible protocol and state what data and uncertainty estimates must be added.

Safe self-study plan. Work with a simulation, for example a pendulum or circuit model, so that you never deal with mains voltage, lasers or high temperatures. Choose one research question, state a hypothesis and list the independent, dependent and controlled variables. Write a protocol: the instrument or simulation settings, resolution, number of trials $N$, and the averaging rule. Run $N \ge 5$ trials at each point, record the raw data in a table, and compute $\bar{x}$ and $u = s/\sqrt{N}$. Then test reproducibility: give the protocol to a peer or repeat the experiment a few days later without changing the protocol text, and compare the results within the stated uncertainty.

Show the structure of a reproducible investigation and the place of each element, from question and variables to the reproducibility check.

Block diagram, left to right: Question $\to$ Hypothesis and theory $\to$ Variables (independent $x$, dependent $y$, controlled $c_i$) $\to$ Protocol (instrument, resolution, $N$, calibration) $\to$ Raw data $\to$ Analysis ($\bar{x}$, $u$) $\to$ Reproducibility check. Below it, a data table with columns $i$, $x_i$, $y_i$, notes, and underneath the formula $u = s/\sqrt{N}$.

A student times a classmate over $100\,\text{m}$ on the same school track. Three runs give $t_1 = 12.4\,\text{s}$, $t_2 = 12.6\,\text{s}$, $t_3 = 12.3\,\text{s}$, with mean $\bar t = 12.43\,\text{s}$. The stopwatch is started when the rear hand leaves the ground and stopped when the chest crosses the finish line; the timer's reaction adds roughly $0.2\,\text{s}$ to every reading. To make the result reproducible, the same starter, the same painted line, and the same stopwatch are used, and every run starts from a standing position after the same countdown.

The scenario exercises operational definitions: 'start' and 'finish' must be fixed by a rule, not by intuition. Repeated trials estimate random scatter around $\bar t$, while the reaction time is a systematic offset that shifts all three values in the same direction. A reproducible investigation states both the protocol and the uncertainty.

$250\,\text{mL}$ of water at $90\,^\circ\text{C}$ are poured into a ceramic mug in a room at $22\,^\circ\text{C}$. Every $60\,\text{s}$ a digital thermometer with resolution $0.5\,^\circ\text{C}$ is read at the same depth: $90$, $82$, $75$, $69$, $64\,^\circ\text{C}$. Reproducibility requires the same mug, the same volume, the same starting temperature, and a note whether the lid is on or off.

Control of variables is the core of the design: volume, container, and room temperature are held constant, so time is the only independent variable and temperature the dependent one. The thermometer's $0.5\,^\circ\text{C}$ resolution sets the measurement uncertainty, and the written protocol lets another student repeat the same curve.

A cyclist rides at $5.0\,\text{m/s}$ on dry asphalt and brakes fully; chalk marks give braking distances $3.2\,\text{m}$, $3.4\,\text{m}$, $3.1\,\text{m}$. Each run uses the same bike, tire pressure $3.0\,\text{bar}$, rider mass $60\,\text{kg}$, and the same braking technique from the moment the lever is pulled to the stop. Air temperature is $25\,^\circ\text{C}$ and wind $1\,\text{m/s}$.

The design isolates speed as the independent variable by fixing mass, tires, and surface; three repeats expose random variation about $3.23\,\text{m}$. This is where theory guides the choice of variables: kinetic energy and friction predict that speed matters, so other factors are controlled rather than varied.