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

D. 0.9

Step-by-step 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