Question

1. 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) 0.60 0.46 0.81 0.55

Explanation:

Step1: Analyze forces on safe

When the safe is on the verge of sliding, the force of static - friction $f_s$ and the component of the gravitational force along the ramp $F_{g\parallel}$ are in balance. The normal force $N$ on the safe on the ramp is $N = mg\cos\theta$, and the component of the gravitational force along the ramp is $F_{g\parallel}=mg\sin\theta$. The maximum static - friction force is $f_s=\mu_sN$, where $\mu_s$ is the coefficient of static friction.

Step2: Set up equilibrium equation

At the angle $\theta$ where the safe just starts to slide, $f_s = F_{g\parallel}$. Substituting $f_s=\mu_sN$ and $N = mg\cos\theta$ into $f_s = F_{g\parallel}$, we get $\mu_smg\cos\theta=mg\sin\theta$.

Step3: Solve for $\mu_s$

Canceling out $mg$ from both sides of the equation $\mu_smg\cos\theta=mg\sin\theta$, we have $\mu_s=\tan\theta$.

Step4: Calculate $\mu_s$

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

Answer:

0.60