Question

a pebble is placed on a ramp. the ramps angle is adjustable. the ramp can be raised up to a maximum angle of 42 degrees before the pebble starts to move. what must be the coefficient of static friction for the pebble?

Explanation:

Step1: Analyze forces on pebble

When the pebble is on the verge of moving, the gravitational force component down the ramp \(mg\sin\theta\) is equal to the maximum static - friction force \(f_s=\mu_sN\). The normal force \(N = mg\cos\theta\) (from perpendicular - to - ramp force balance).

Step2: Equate forces and solve for \(\mu_s\)

Since \(mg\sin\theta=\mu_smg\cos\theta\), we can cancel out \(mg\) (assuming \(m
eq0\)). Then \(\mu_s=\frac{\sin\theta}{\cos\theta}=\tan\theta\).

Step3: Substitute \(\theta = 42^{\circ}\)

\(\mu_s=\tan(42^{\circ})\). Using a calculator, \(\tan(42^{\circ})\approx0.90\)

Answer:

\(0.90\)