Answer:
θ = 8.16º and μ = 0.0788
Explanation:
This is a case of a curve with an angle of inclination, cant, to help the cars to take which, in a free body diagram, see that the normal is the force that we must decompose.
Let's start by reducing the speeds to the SI system
v1 = 63 km / h (1000m / 1km) (1h / 3600s) = 17.5 m / s
v2 = 42 km / h (1000m / 1km) (1h / 3600s) = 11.67 m / s
Let us write Newton's second law, for this type of case the located references the horizontal X axis and the vertical Y axis, so the only force that we must decompose is the Normal (N)
Axis y
Ny -W = 0
Ny = W
X axis
Nx = m a
a = v² / R
Let's use trigonometry to break down the force
sin θ = Nx / N
Nx = N sin θ
cos θ = Ny / N
Ny = N cos θ
We replace and calculate the angle of the cant
N cos θ = W = mg
N = mg / cos θ
N sin θ = m v² / R
(mg / cos θ) sin θ = m v² / R
tan θ = v² / gR (1)
θ = tan-1 (v² / gR)
θ = tan-1 (17.5² / 9.8 218)
θ = tan-1 (0.143)
θ = 8.16º
This is the angle of the curve that is constant
If the car goes this speed the friction force is zero, for different speeds this force begins to appear, in the second case the car goes slower, so the normal component would take it towards the center of the curve, whereby a force twelve appears that opposes this movement, consequently, the force of friction and directs towards the outside of the curve
We write Newton's law
Nx - fr = m a
The expression for force rubbing is
fr = μ N
N sin θ - μ N = mv² / R
N (sin θ - μ) = m v² / R
(mg / cos θ) (sin θ -μ) = (m v² / R)
sin θ - μ = v² cos θ / gR
μ = sin θ - v² cos θ / gR
Calculate
μ = sin 8.16 - 11.67²2 cos 8.16 / 9.8 218
μ = 0.1419 - 0.0631
.
μ = 0.0788
