Properties of Friction — Question 9

Question 9

An area test can falsify an overextended friction model

Take $g=10$ m/s$^2$. Use ideal dry friction, $|f_s|\le\mu_sN$ and $|f_k|=\mu_kN$. A 3 kg rectangular block slides on the same level material in two orientations. Contact areas are 20 and 40 cm$^2$. With no pull, measured decelerations are respectively 2 and 3 m/s$^2$. Assume measurement error is negligible and no other horizontal forces act.

1. Infer the kinetic coefficient in each trial.

2. Can one constant area-independent coefficient fit both trials?

3. For a new 6 kg block, what predictions follow from the two fitted coefficients, and why is neither guaranteed by these data?

Question 9 - Solution

An area test can falsify an overextended friction model

(a) Infer from force and normal load.

For unforced sliding, the deceleration magnitude is $$m|a|=\mu_kmg\Longrightarrow\mu_k=\frac{|a|}{g}.$$ $$\boxed{\mu_{k,20}=\frac2{10}=.20},\qquad\boxed{\mu_{k,40}=\frac3{10}=.30}.$$

(b) Compare with the model’s restriction.

One common coefficient would require $$|a_{20}|=\mu_kg=|a_{40}|.$$ The measured $2\ne3$ m/s$^2$ contradicts that restricted model. The comparison does not isolate contact area as the cause: changing orientation may also change other surface properties.

(c) State conditional extrapolations.

Conditional on transferring either fitted coefficient to the new block, $$|a|=\frac{\mu_k(6)g}{6}=\mu_kg =\begin{cases}2\ \mathrm{m/s^2},&\mu_k=.20,\\3\ \mathrm{m/s^2},&\mu_k=.30.\end{cases}$$ Mass cancels within each ideal model. The two measurements establish neither coefficient’s applicability to the new contact.