A flywheel turns on its axle, a weight hangs from a string and falls one metre. A stopwatch gives you the time, and the lab manual says plot it, take the slope, and there is your moment of inertia. The plot fits beautifully – an R² above 0.999 – and the answer is several percent wrong.
In this experiment you drop six different masses, each several times, and then you look at the thing almost nobody looks at: the residuals. Those tiny distances from each point to the line, which the R² crushes into a single number and makes disappear.
Why this experiment is different
In the spring constant two ways we saw data that was structurally incapable of revealing its own error – the only escape was to leave the experiment. Here the opposite happens: the evidence was inside those six points all along, and it is the procedure that throws it away.
Write the signs of the residuals in a row and a pattern appears: + − − − + +, a clean U. The points are not scattered randomly about the line; the line is cutting through a curve. The model is wrong, and the data knew it from the start.
The sharpest moment on the page is realising that both numbers came out of one single fit. One was the R² and it told you the fit was excellent. The other was the intercept, which should have been 1/g, and it told you that gravity points upwards. The fit does not report the contradiction, because spotting contradictions is not its job – its job is to find the best line, and it will find the best line through meaningless data too.
The escape is small: find the lightest rider that keeps the wheel turning steadily. That one thirty-second measurement gives you the friction torque, and once the correction is on, the point-by-point moment of inertia stops drifting with mass. As in density by Archimedes, the difference between a good measurement and a correct one is usually that one extra step.
In the Python section we build four thousand synthetic data sets that are straight lines by construction and carry your own measured noise, and we see that the R² you admired is about ten sigma worse than a genuine line would have scored. Then we hand the same six untouched points to the correct model and recover both the moment of inertia and the friction torque – without a single extra measurement.
The rule worth taking away: a fit statistic never means anything on its own, only against the value you should have got.