As a civil engineering intern during one of your summers in college, you are asked to design a curved section of roadway that meets the following conditions: When ice is on the road, and the coefficient of static friction between the road and rubber is 0.077, a car at rest must not slide into the ditch and a car traveling less than 63 km/h must not skid to the outside of the curve. Neglect the effects of air drag and rolling friction. What is the minimum radius of curvature of the curve and at what angle should the road be banked?
To find the minimum radius of curvature and angle of banking for the curved section of roadway, we need to analyze the forces acting on the car under the given conditions.
First, let's consider the car at rest on the icy road. In this case, the car must not slide into the ditch, which means the frictional force acting on the car should be equal to or greater than the force trying to make it slide towards the ditch. The force trying to make the car slide is the component of the car's weight acting parallel to the road surface.
The equation for the maximum static frictional force (Fs) is given by Fs = coefficient of static friction (μ) * normal force (N). In this case, the normal force will be equal to the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity.
Now, let's consider the car traveling at a speed less than 63 km/h. In this case, the car must not skid to the outside of the curve, which means the frictional force acting on the car should be equal to or greater than the force trying to make it slide away from the center of the curve. The force trying to make the car slide is the centripetal force (Fc), which is given by Fc = (mv^2) / r, where m is the mass of the car, v is the velocity of the car, and r is the radius of curvature of the road.
To prevent the car from sliding, we can equate the maximum static frictional force to the centripetal force:
μN = (mv^2) / r
Since we have "m" on both sides of the equation, we can cancel it out:
μg = (v^2) / r
Now, let's plug in the values and solve for the minimum radius of curvature (r):
0.077 * g = (63 km/h)^2 / r
Note: To convert km/h to m/s, divide by 3.6.
0.077 * g = (63 km/h)^2 / (r * (3.6 m/s / 1 km/h)^2)
Simplifying the equation further:
0.077 * g = (63^2 * (1 km/h)^2) / (r * (3.6 m/s)^2)
0.077 * 9.8 m/s^2 = (63^2 * 1^2) / (r * 3.6^2 m/s^2)
Now, we can solve for the minimum radius of curvature (r):
r = (63^2 * 1^2) / (0.077 * 9.8 * 3.6^2)
Using a calculator, we find that the minimum radius of curvature is approximately 154.47 meters.
Now, let's calculate the angle at which the road should be banked. For a car to safely navigate the curve without skidding, we need to introduce a banking angle (θ). The banking angle allows for a horizontal component of the normal force to provide the required centripetal force.
The formula for the banking angle (θ) can be given as:
θ = arctan((v^2) / (r * g))
Plugging in the values, we have:
θ = arctan((63 km/h)^2 / (154.47 m * 9.8 m/s^2))
θ = arctan(((63 / 3.6 m/s)^2) / (154.47 * 9.8))
Again, using a calculator to solve the equation, we find that the angle of banking (θ) is approximately 3.26 degrees.
Therefore, the minimum radius of curvature of the curve should be approximately 154.47 meters, and the road should be banked at an angle of approximately 3.26 degrees.
use the equation tan0= mu s
mu s being the coeffictient of static friction