Question

Some car manufacturers claim that their vehicles could climb a slope of 42 ∘. For this to be possible, what must be the minimum coefficient of static friction between the vehicle’s tires and the road?
A. 0.4
B. 0.5
C. 0.7
D. 0.9

Answer 1

Answer:

D. 0.9

Explanation:

Calculating minimum coefficient of static friction, we first resolve the forces (normal and frictional) acting on the vehicle at an angle to the horizontal into their x and y components. After this, we can now substitute the values of x and y components into equation of static friction. Diagrammatic illustration is attached.

Resolving into x component:

                        ∑$F_{x} = F_{s} - mgsin\alpha =0$

                          $F_{s} = mgsin\alpha$     ------(1)

Resolving into y component:

                        ∑$F_{y} = F_{n} - mgcos\alpha =0$

                          $F_{n} = mgcos\alpha$      ------(2)

Static frictional force, $F_{s} \leq$ μ $F_{n}$       ------(3)

substituting $F_{s}$ from equation (1) and $F_{n}$ from equation (2) into equation (3)

                         $mgsin\alpha \leq$ μ $mgcos\alpha$

                         $sin\alpha \leq$ μ $cos\alpha$

                         μ $\geq \frac {sin\alpha}{cos\alpha}$

                         μ $\geq tan\alpha$

The angle the vehicles make with the horizontal α = 42°

                         μ ≥ tan 42°

                         μ ≥ 0.9