A circular curve of highway is designed for traffic moving at 93 km/h. Assume the traffic consists of cars without negative lift. (a) If the radius of the curve is 110 m, what is the correct angle of banking of the road? (b) If the curve were not banked, what would be the minimum coefficient of friction between tires and road that would keep traffic from skidding out of the turn when traveling at 93 km/h?
To answer both parts of the question, we need to use the principle of centripetal force. Centripetal force is the force that keeps an object moving in a circular path. In the case of a car on a curved road, the centripetal force is provided by the friction between the tires and the road.
(a) To determine the correct angle of banking of the road, we need to consider the centripetal force acting on the car. The centripetal force is given by the equation:
F = mv^2 / r
Where:
F = Centripetal force
m = Mass of the car
v = Velocity of the car
r = Radius of the curve
In this case, we are given the velocity, v, which is 93 km/h. However, we need to convert it to m/s by dividing by 3.6.
93 km/h ÷ 3.6 = 25.833 m/s (rounded to three decimal places)
The radius of the curve, r, is given as 110 m.
Now, we can solve for the angle of banking, θ. The angle of banking is related to the tangent of the angle, which can be defined as:
tan(θ) = v^2 / (g * r)
Where:
g = Acceleration due to gravity (9.8 m/s^2)
Plugging in the known values:
tan(θ) = (25.833 m/s)^2 / (9.8 m/s^2 * 110 m)
Now, calculate the numerator:
(25.833 m/s)^2 = 667.194489 m^2/s^2
Calculating the denominator:
9.8 m/s^2 * 110 m = 1078 m^2/s^2
Substituting these values into the equation:
tan(θ) = 667.194489 m^2/s^2 / 1078 m^2/s^2
Using an inverse tangent function, find the angle, θ:
θ = arctan(667.194489 m^2/s^2 / 1078 m^2/s^2)
Calculating:
θ ≈ 30.844 degrees (rounded to three decimal places)
Therefore, the correct angle of banking for the road is approximately 30.844 degrees.
(b) Now, let's determine the minimum coefficient of friction required to prevent skidding when the curve is not banked.
The maximum force of static friction (Fs) can be defined as:
Fs = μs * N
Where:
Fs = Force of static friction
μs = Coefficient of static friction
N = Normal force
In this case, the normal force is equal to the weight of the car, which can be found using:
N = mg
Where:
m = Mass of the car
g = Acceleration due to gravity (9.8 m/s^2)
Let's substitute the known values into the equation:
N = m * 9.8 m/s^2
Now, we need to calculate the mass of the car. Unfortunately, we don't have information about the mass of the car. Assuming the mass of an average car is around 1500 kg, let's continue with that estimation.
N = 1500 kg * 9.8 m/s^2
N = 14,700 N
Now, let's calculate the centripetal force, using the given velocity and radius:
F = mv^2 / r
F = 1500 kg * (25.833 m/s)^2 / 110 m
F ≈ 9,760 N (rounded to the nearest whole number)
Since Fs must provide this centripetal force, we can set Fs equal to F:
Fs = F
μs * N = 9,760 N
μs * 14,700 N = 9,760 N
Simplifying:
μs ≈ 0.664 (rounded to three decimal places)
Therefore, the minimum coefficient of friction required to prevent skidding when the curve is not banked is approximately 0.664.