A banked circular highway curve is designed for traffic moving at 60 km/h. The radius of the curve is 205 m. Traffic is moving along the highway at 40 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the turn without sliding off the road?
Somehow the word "banked" got typed as "naked". I must not be quite awake yet
To find the minimum coefficient of friction between tires and road that will allow cars to negotiate the turn without sliding off the road, we can use the formula for the minimum coefficient of friction in a banked curve:
μ ≥ tan(θ)
Where:
μ = coefficient of friction
θ = angle of banking of the curve
First, let's calculate the angle of banking of the curve:
tan(θ) = v^2 / (g * r)
Where:
v = velocity of the car (in m/s)
g = acceleration due to gravity (usually taken as 9.8 m/s²)
r = radius of the curve
Converting the given velocities to m/s:
v = 40 km/h = (40 * 1000) / 3600 = 11.11 m/s
Now, let's calculate the angle of banking:
tan(θ) = (11.11^2) / (9.8 * 205)
θ = arctan((11.11^2) / (9.8 * 205))
θ ≈ 0.348 radians
Finally, let's calculate the minimum coefficient of friction:
μ ≥ tan(θ)
μ ≥ tan(0.348)
μ ≈ 0.368
Therefore, the minimum coefficient of friction between tires and road that will allow cars to negotiate the turn without sliding off the road is approximately 0.368.
To determine the minimum coefficient of friction required for cars to negotiate the turn without sliding off the road, we can use the concept of centripetal force. The centripetal force is the force that keeps an object moving in a curved path.
The formula for centripetal force is:
F = (m * v^2) / r
Where:
F is the centripetal force,
m is the mass of the car,
v is the velocity of the car, and
r is the radius of the curve.
First, let's convert the given velocities to meters per second:
60 km/h = (60 * 1000) / 3600 = 16.67 m/s (velocity on the banked curve)
40 km/h = (40 * 1000) / 3600 = 11.11 m/s (velocity of the traffic)
Next, we need to determine the effective gravitational force acting on the car. On a banked curve, there are two components of gravitational force: the vertical component (mg * cosθ) and the horizontal component (mg * sinθ). Since the car is not sliding off the road, these two forces must be in equilibrium. Therefore, the horizontal component provides the necessary centripetal force.
mg * sinθ = (m * v^2) / r
Next, we substitute the numerical values:
(205 * 9.8) * sinθ = (m * 11.11^2) / 205
Simplifying the equation:
0.2 * sinθ = (0.05 * m)
Now, we need to eliminate the variable m, which is the mass of the car. To do that, let's consider the normal force acting on the car (mg * cosθ) and substitute it into our equation. The normal force is equal to the gravitational force acting on the car:
0.2 * sinθ = (0.05 * (mg * cosθ))
Now, we can cancel out the mass from both sides of the equation:
0.2 * sinθ = (0.05 * g * cosθ)
Next, we can simplify the equation further by dividing both sides by 0.05:
4 * sinθ = (g * cosθ)
Finally, we can determine the minimum coefficient of friction (μ) using the relationship between the cosine and sine functions:
μ = tanθ
In other words, the minimum coefficient of friction required is equal to the tangent of the angle of inclination of the banked curve. Therefore, the answer to the question is:
The minimum coefficient of friction required for cars to negotiate the turn without sliding off the road is equal to the tangent of the angle of inclination of the banked curve.
Use the 60 km/h "naked curve design speed" to determine th banking angle, A. First convert the speed to m/s.
(M V^2/R) = M g sin A
sin A = V^2/(Rg)
If the car is travelling at a slower speed, the friction force plus the gravity component must exceed the centripetal force in the radial direction along the surface. Use that fact to determine the required friction coefficient.