Question

a box is initially held at rest on a ramp with an incline of 37° with respect to the horizontal and then released. the coefficients of static and kinetic friction between the box and the ramp are 0.50 and 0.20, respectively. which of the following best describes the motion, if any, of the box after it is released?
a the box will remain at rest on the incline.
b the box will start sliding down the incline at a constant velocity.
c the box will start sliding down the incline and then slow down at a constant rate.
d the box will start sliding down the incline and then continue to speed up at a constant rate.

Explanation:

Step1: Calculate the maximum static - friction force

The formula for the maximum static - friction force is \(f_{s,\text{max}}=\mu_{s}N\). On an incline, \(N = mg\cos\theta\), so \(f_{s,\text{max}}=\mu_{s}mg\cos\theta\). Given \(\mu_{s}=0.50\) and \(\theta = 37^{\circ}\), \(f_{s,\text{max}}=0.50mg\cos37^{\circ}\approx0.50mg\times0.8 = 0.4mg\).

Step2: Calculate the gravitational force along the incline

The gravitational force along the incline is \(F_{g\parallel}=mg\sin\theta\). With \(\theta = 37^{\circ}\), \(F_{g\parallel}=mg\sin37^{\circ}\approx mg\times0.6=0.6mg\).

Step3: Compare \(F_{g\parallel}\) and \(f_{s,\text{max}}\)

Since \(F_{g\parallel}(0.6mg)>f_{s,\text{max}}(0.4mg)\), the box will start to move.

Step4: Calculate the kinetic - friction force

The kinetic - friction force is \(f_{k}=\mu_{k}N=\mu_{k}mg\cos\theta\). Given \(\mu_{k}=0.20\) and \(\theta = 37^{\circ}\), \(f_{k}=0.20mg\cos37^{\circ}\approx0.20mg\times0.8 = 0.16mg\).

Step5: Calculate the net force along the incline

The net force along the incline is \(F_{\text{net}}=F_{g\parallel}-f_{k}\). Substituting the values, \(F_{\text{net}}=0.6mg - 0.16mg=0.44mg\).

Step6: Determine the acceleration

Using Newton's second law \(F = ma\), \(a=\frac{F_{\text{net}}}{m}\). Since \(F_{\text{net}} = 0.44mg\), \(a = 0.44g\). The acceleration is constant (\(a>0\)), so the box will speed up at a constant rate.

Answer:

D. The box will start sliding down the incline and then continue to speed up at a constant rate.