Imagine that you are designing a circular curve in a highway that must have a radius of 330 ft and will carry traffic moving at 60 mi/h.
a) At what angle should the roadbed be banked for maximum safety?
b) If the roadway is not banked, what would the necessary coefficient of static friction between the tires and the asphalt road have to be to keep a car on the road? Is such a coefficient reasonable? (Hint: 1 m=3.3 ft, 1 m/s=2.24 mi/h.)
To answer these questions, we need to consider the forces acting on a car as it moves along the curved section of the highway. There are two key forces involved: the gravitational force acting vertically downwards and the frictional force acting horizontally towards the center of the curve.
a) To determine the angle at which the roadbed should be banked for maximum safety, we need to assess the balance between these forces. At the maximum safe angle, the resultant of these forces would act directly downwards, perpendicular to the road surface. This means that the frictional force would be completely utilized in helping the car to turn, while the gravitational force would not cause the car to skid or slide off the road.
To find this angle, we can use the following equation:
tan(theta) = v^2 / (g * r)
where:
- theta is the angle of banking
- v is the velocity of the car (60 mi/h converted to ft/s)
- g is the acceleration due to gravity (32.2 ft/s^2)
- r is the radius of the curve (330 ft)
First, let's convert the car's velocity from miles per hour (mi/h) to feet per second (ft/s):
60 mi/h * (5280 ft/mi) / (60 min/h * 60 s/min) = 88 ft/s
Now we can substitute the values into the formula:
tan(theta) = (88 ft/s)^2 / (32.2 ft/s^2 * 330 ft)
Using a scientific calculator, we can calculate the value of theta by taking the inverse tangent (arctan) of both sides of the equation:
theta = arctan((88 ft/s)^2 / (32.2 ft/s^2 * 330 ft))
This will give us the angle of banking required for maximum safety.
b) If the roadway is not banked, we need to determine the necessary coefficient of static friction between the tires and the asphalt road to keep the car on the road. In this case, the frictional force is solely responsible for providing the centripetal force required to keep the car moving in a circle.
The equation for the maximum frictional force (F_friction_max) is:
F_friction_max = μ * N
where:
- μ is the coefficient of static friction
- N is the normal force (equal to the weight of the car)
The normal force is given by:
N = m * g
where:
- m is the mass of the car
- g is the acceleration due to gravity (32.2 ft/s^2)
The centripetal force (F_centripetal) is given by:
F_centripetal = m * (v^2 / r)
To keep the car on the road, the maximum frictional force must equal the centripetal force:
F_friction_max = F_centripetal
Substituting the equations for the maximum frictional force and the centripetal force, we get:
μ * N = m * (v^2 / r)
Replacing N with m * g, we have:
μ * m * g = m * (v^2 / r)
We can see that the mass (m) cancels out, simplifying the equation to:
μ * g = v^2 / r
Now, solving for μ:
μ = (v^2 / r) / g
Substituting the given values:
μ = ((88 ft/s)^2 / 330 ft) / 32.2 ft/s^2
Calculate this expression to find the coefficient of static friction required to keep the car on the road.
Finally, we can assess the reasonability of this coefficient by comparing it to typical values for the static friction between tires and asphalt.