Designing a Safe Curved Roadway: AP Physics Problem for Civil Engineers

[daniel-c]

A civil engineer is asked to design a curved section of roadway that meets the following conditions: With ice on the road, when the coefficient of static friction between the road and rubber is 0.08, a car at rest must not slide into the ditch and a car traveling less than 50 km/h must not skid to the outside of the curve. What is the minimum radius of curvature of the curve and at what angle should the road be banked?

 

[Dick]

Any thoughts on this? What forces acting on the car when it's at rest? What changes when it's in motion (rotational acceleration?). The rules of the forum are that you have to start trying to solve the problem before people can jump in and help.

 

[Moetasim]

I would approach this problem by first understanding the physics principles involved. The coefficient of static friction is a measure of the friction force between two surfaces when they are not moving relative to each other. In this case, the two surfaces are the road and the rubber tires of a car. The coefficient of static friction is affected by factors such as the materials involved, the weight of the car, and the angle of the road.

To design a safe curved roadway, the minimum radius of curvature and the angle of banking must be determined. The minimum radius of curvature can be calculated using the formula v^2/r = μ, where v is the speed of the car, r is the radius of curvature, and μ is the coefficient of static friction. Since the speed of the car is given as 50 km/h, which is equivalent to 13.9 m/s, and the coefficient of static friction is 0.08, we can solve for the minimum radius of curvature to be 194.4 m.

Next, the angle of banking can be determined using the formula tanθ = v^2/(rg), where θ is the angle of banking, v is the speed of the car, r is the radius of curvature, and g is the acceleration due to gravity. Plugging in the values, we get a required angle of banking of approximately 3.5 degrees.

In order to ensure the safety of the curved roadway, the civil engineer should also consider other factors such as the weight of the cars that will be using the road and the condition of the road surface. They should also regularly monitor and maintain the road to ensure that it meets safety standards.

In conclusion, designing a safe curved roadway for civil engineers involves understanding the physics principles involved and using mathematical calculations to determine the minimum radius of curvature and angle of banking. It is important for engineers to consider other factors and regularly maintain the road to ensure the safety of drivers.