Question

a safe rests on a boat ramp. the ramp is tilted until the safe just starts to slide. the angle at which the safe starts to slide is 31°. find the coefficient of static friction between the safe and the ramp. (free body diagram is in equilibrium)

Explanation:

Step1: Analyze forces on the safe

When the safe is on the verge of sliding, the force of static - friction $f_s$ is at its maximum value $f_{s,max}=\mu_sN$, and the forces along the ramp and perpendicular to the ramp are in equilibrium. Let the mass of the safe be $m$, the angle of the ramp be $\theta = 31^{\circ}$, the normal force be $N$, and the gravitational force be $mg$. In the direction perpendicular to the ramp, $N - mg\cos\theta=0$, so $N = mg\cos\theta$. In the direction along the ramp, $mg\sin\theta=f_{s,max}=\mu_sN$.

Step2: Substitute $N$ into the friction equation

Substitute $N = mg\cos\theta$ into $mg\sin\theta=\mu_sN$. We get $mg\sin\theta=\mu_smg\cos\theta$. Then $\mu_s=\tan\theta$.

Step3: Calculate the coefficient of static friction

Given $\theta = 31^{\circ}$, $\mu_s=\tan(31^{\circ})\approx0.60$.

Answer:

$0.60$