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)

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 $F_{s,max}=\mu_sN$, where $\mu_s$ is the coefficient of static friction and $N$ is the normal force. Resolve the weight of the safe $mg$ into components. The component of the weight along the ramp is $mg\sin\theta$ and the component perpendicular to the ramp is $mg\cos\theta$, where $\theta = 31^{\circ}$ is the angle of the ramp. In the direction perpendicular to the ramp, the normal force $N$ and the perpendicular - component of the weight are in equilibrium, so $N = mg\cos\theta$. In the direction along the ramp, at the verge of sliding, $F_{s,max}=mg\sin\theta$.

Step2: Substitute $F_{s,max}=\mu_sN$ into the equilibrium equation

Since $F_{s,max}=mg\sin\theta$ and $F_{s,max}=\mu_sN$ and $N = mg\cos\theta$, we have $\mu_smg\cos\theta=mg\sin\theta$. The mass $m$ and the acceleration due to gravity $g$ cancel out, and we get $\mu_s=\tan\theta$.

Step3: Calculate the coefficient of static friction

Substitute $\theta = 31^{\circ}$ into the formula $\mu_s=\tan\theta$. Using a calculator, $\tan(31^{\circ})\approx0.60$.

Answer:

0.60