Properties of Friction — Question 3

Question 3

Onset and steady sliding measure different coefficients

Take $g=10$ m/s$^2$. Use ideal dry friction, $|f_s|\le\mu_sN$ and $|f_k|=\mu_kN$. A block on an adjustable incline begins to slide when the angle is increased just beyond $30^\circ$. During a separate steady downward sliding test, it moves at constant speed at angle $20^\circ$. Assume the same coefficients in both tests.

1. Infer $\mu_s$ and $\mu_k$.

2. If the already-sliding block is on a $25^\circ$ incline, find its acceleration.

3. Could a block initially at rest remain at rest at $25^\circ$? Explain the history dependence.

Question 3 - Solution

Onset and steady sliding measure different coefficients

(a) Use the correct equality for each test.

At impending slip and at steady sliding, respectively, $$mg\sin30^\circ=\mu_smg\cos30^\circ\Longrightarrow\boxed{\mu_s=\tan30^\circ=.57735},$$ $$mg\sin20^\circ=\mu_kmg\cos20^\circ\Longrightarrow\boxed{\mu_k=\tan20^\circ=.36397}.$$

(b) Use kinetic friction for the moving block.

For a block already moving downhill, use kinetic friction: $$a=g(\sin25^\circ-\mu_k\cos25^\circ).$$ $$\boxed{a=10[\sin25^\circ-(\tan20^\circ)\cos25^\circ]=.9275\ \mathrm{m/s^2}\text{ downhill}}.$$

(c) Test the static inequality independently.

A resting block instead requires $$mg\sin25^\circ\leq\mu_smg\cos25^\circ \Longleftrightarrow\tan25^\circ=.4663<.57735.$$ Rest is feasible. At the same angle, an already sliding block accelerates because the kinetic coefficient is smaller; the contact regime depends on the motion history.