Question

A flatbed truck is carrying a 20-kg crate up a sloping road. The coefficient of static friction between the create and the bed is 0.40, and the coefficient of kinetic friction is 0.30. Draw a free body diagram for the crate when the truck is one the sloping road. What is the maximum angle of the slope that the truck can climb at constant speed so the crate doesn't slip?

Answer 1

θ = 21.8º

Step-by-step explanation:

We can solve this exercise using Newton's conditions for equilibrium, in the attached we can see a diagram of the forces.

The most used coordinate system is an axis parallel to the plane (x axis) and an axis perpendicular to the plane (y axis), let's write Newton's equations on this axes

Y Axis

N-
`W_(y)` = 0

N =
`W_(y)`

X axis

fr - Wₓ = 0 (1)

Let's use trigonometry to find the normal ones

sin θ = Wₓ / W

cos θ =
`W_(y)` / W

Wx = W sin θ

`W_(y)` = W cos θ

The friction force has the formula

fr = μ N

fr = μ (W cos θ)

We substitute in 1

μ mg cosθ = mg sin θ

μ cos θ = sin θ

tan θ = μ

θ = tan⁻¹ μ

calculate

θ = tan⁻¹ 0.40

θ = 21.8º