Question

a small package is placed on a ramp. the package has a coefficient of static friction equal to 0.32. what is the maximum angle that the ramp can have before the package will start to slide down the ramp in degrees? answer

Explanation:

Step1: Analyze the forces

When the package is on the verge of sliding, the force of static friction \(f_s\) balances the component of gravitational force along the ramp. The normal force \(N = mg\cos\theta\) and the component of gravitational force along the ramp is \(mg\sin\theta\). The maximum static - friction force \(f_s=\mu_sN\). So, \(mg\sin\theta=\mu_smg\cos\theta\).

Step2: Solve for \(\theta\)

Cancel out \(mg\) from both sides of the equation \(mg\sin\theta=\mu_smg\cos\theta\). We get \(\tan\theta=\mu_s\). Given \(\mu_s = 0.32\), then \(\theta=\arctan(\mu_s)\).

Step3: Calculate the value of \(\theta\)

Using a calculator, \(\theta=\arctan(0.32)\approx17.7^{\circ}\)

Answer:

\(17.7^{\circ}\)