# Uncertainty of the coefficient of friction

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?

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.