Question

a box is placed on a usps trucks ramp and starts to slide down the ramp. the box has a mass of 21 kilograms and a coefficient of kinetic friction equal to 0.31. the ramp is at an angle of 40 degrees. calculate the acceleration of the box in meters per second per second.

Explanation:

Step1: Analyze forces

The force along the ramp due to gravity is \(F_{g}=mg\sin\theta\), and the frictional force is \(F_{f}=\mu_{k}mg\cos\theta\). By Newton's second law \(F = ma\), the net force \(F_{net}=mg\sin\theta-\mu_{k}mg\cos\theta\).

Step2: Solve for acceleration

Divide the net - force equation by \(m\) (since \(F_{net} = ma\)). So \(a = g\sin\theta-\mu_{k}g\cos\theta\). Given \(g = 9.8\ m/s^{2}\), \(\theta = 40^{\circ}\), \(\mu_{k}=0.31\).
\(\sin40^{\circ}\approx0.643\), \(\cos40^{\circ}\approx0.766\)
\(a=9.8\times0.643 - 0.31\times9.8\times0.766\)
\(a = 9.8\times(0.643-0.31\times0.766)\)
\(a = 9.8\times(0.643 - 0.237)\)
\(a=9.8\times0.406\)

Answer:

\(a\approx3.98\ m/s^{2}\)