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We all know the experiment in which one raises the angle of an inclined plane until right before the block begins to slide, in order to establish the static coefficient of friction.

The result for static coefficient of friction is then $\mu_s = \tan{(\theta)}$, where $\theta$ is this critical angle.

In laboratory it is usual to repeat this for ten times, and then the uncertainty is the standard deviation.

My question is: If I have only one trial is it fine to estimate the uncertainty from $\sec^2(\theta)$? If the uncertainty in angle is $\pm 1^\circ = \frac{\pi}{180}$ radians (because it is a digital level measurement), then the uncertainty in tangent is $|\sec^2(\theta)| \frac{\pi}{180}$. For instance, if $\theta = 15^\circ$, this would give $0.27 \pm 0.02$.

Is that correct? Is it legit to estimate the uncertainty in one trial knowing the uncertainty in measured angle?

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You do not only describe two scenarios in your questions, but the two errors really differ in their nature:

  1. If you repeatedly measure a quantity, the standard model is that the measured value $y_i$ consists of the true value $y_{true}$ and a random error $\epsilon_i$ -- often assumed to be normally distributed with a mean value zero and a standard deviation $\sigma_\epsilon$. The basic idea of this model is that each time we take a new measurement the random error is responsible for the difference in our results. Thus, if the surface roughness of the object or the inclined plane is not perfectly homogeneous, the error is accounted for by $\epsilon_i$.

  2. In contrast, if you take a single measurement and argue that the precision of the gage (used to measure the angle) is $\sigma_u = 1deg$, you have not included the inhomogeneous of the surface roughness.

Of course, if you conduct your experiment in a well controlled environment, where the humidity of the air, the surface roughness of the incline plane, the speed in which the angle is changed etc. have a negligible uncertainty, you could argue that the dominant contribution to the error stems from the precision of gage. In this case your single measurement uncertainty is valid. However, I reckon it is good practice to take several measurements, because ultimately, we do not know all the influences and/or we always should account mistakes -- e.g. somebody made a scratch of the surface of the inclined plane and we place the object right on top of it.

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  • $\begingroup$ Thank you very much, Semoi. Your answer bring light to my questions about error. I have one more related: what if I have like ten measures from ten similar devices (but not the same)? May I consider the average and the standard error as an answer to the problem? Thank you again. $\endgroup$
    – baseaxis
    Jul 13, 2022 at 1:33
  • $\begingroup$ If all devices have the same precision and accuracy, it is statistically fine to consider the ten measured values as the error distribution. However, if you have not check this assumption of equal precision and accuracy, the argument is no longer solid. Finally, if you know that the precisions and accuracies differ you definitely should use Baysian methods to obtain the most probable answer. $\endgroup$
    – Semoi
    Jul 20, 2022 at 14:19

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