= Inclined-plane measurement of static friction
{title2=$\mu_s=\tan\theta_c$}
A university laboratory places an <aluminium> block on a <Teflon>-lined wooden board. For its <inclined plane> stage, the pulley and block string are removed; a rope and hook connect the board to a crank. The block is placed gently near the sheet's middle without prior sliding. Smooth cranking raises the board; a $0^\circ$–$45^\circ$ <protractor> reads the smallest <angle> at which sliding begins. Four readings are taken without added load, then four each with $200$, $400$, and $800\,\mathrm g$ added symmetrically at two screws. Removing the string isolates gravity-driven onset; gentle placement and smooth raising avoid premature disturbance.<Friction (Physics 183, Experiment 5, Fall 2004)>[Printed p. 5, Part III, steps 1–7 and Figure 6; worksheet p. 2, Table 3]{ref}
Infer $\mu_s=\tan\theta_c$ under the assumptions of <static friction on an inclined plane>. Repetition tests onset scatter; added loads test coefficient constancy. No measured onset values are supplied here.
For a replication, additionally check the <protractor> zero with a level board, retain the same contact face and surface region, and observe whether tipping or rolling precedes sliding. These are proposed additional controls. A finite <angle> resolution and the disturbance needed to detect first motion limit the estimate; a slow ramp does not by itself establish perfectly rigid, dry, or direction-independent contact. At the model's $\mu_s=0.40$ threshold, differentiating the <tangent function> gives $d\mu_s/d\theta=1+\mu_s^2=1.16$, with $\theta$ in <radians>. A hypothetical $0.5^\circ$ <measurement error> would therefore correspond to approximately $0.0101$ in the inferred coefficient at first order. This sensitivity calculation is not an uncertainty reported by the laboratory.
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