A banked circular highway curve is designed for traffic moving at 84 km/h. The radius of the curve is 152 m. Traffic is moving along the highway at 43 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?
You have three forces: gravity, friction, and centripetal force.
Break gravity on the car into two components, normal to the surface, and down the surface (parallel).
Now, break centripetal force in to normal, and up the surface (parallel)
Add the up the surfaces (one is negative, weight), and set them equal to friction (mu*sum of parallel forces).
To find the minimum coefficient of friction, we can start by calculating the maximum speed the cars can safely travel around the curve without sliding off the road. This maximum speed is known as the "design speed" and is given as 84 km/h.
Step 1: Convert the design speed to meters per second.
To convert km/h to m/s, divide by 3.6:
Design speed = 84 km/h ÷ 3.6 = 23.33 m/s (rounded to two decimal places)
Step 2: Calculate the maximum speed the cars can travel on a rainy day.
In this scenario, the traffic is moving at a speed of 43 km/h. We need to convert this speed to meters per second as well:
Speed on a rainy day = 43 km/h ÷ 3.6 = 11.94 m/s (rounded to two decimal places)
Step 3: Calculate the centripetal acceleration required for a curve with the given radius.
Centripetal acceleration (ac) is given by the equation:
ac = v^2 / r
Where v is the velocity and r is the radius of the curve.
For the design speed:
ac_design = (23.33 m/s)^2 / 152 m = 3.57 m/s^2 (rounded to two decimal places)
For the speed on a rainy day:
ac_rainy_day = (11.94 m/s)^2 / 152 m = 0.94 m/s^2 (rounded to two decimal places)
Step 4: Calculate the maximum frictional force available.
The maximum frictional force (F_friction_max) is given by the equation:
F_friction_max = m * g * μ
Where m is the mass of the car, g is the acceleration due to gravity (9.8 m/s^2), and μ is the coefficient of friction.
To find the maximum frictional force, we need the mass of the car. Let's assume it's 1000 kg (this value can vary depending on the type of car).
F_friction_max = 1000 kg * 9.8 m/s^2 * μ
Step 5: Set up the equations for the two scenarios and solve for the coefficient of friction.
For the design speed:
F_friction_design = m * ac_design
1000 kg * 3.57 m/s^2 = 1000 kg * 9.8 m/s^2 * μ_design
μ_design = 3.57 m/s^2 / 9.8 m/s^2 ≈ 0.37
For the speed on a rainy day:
F_friction_rainy_day = m * ac_rainy_day
1000 kg * 0.94 m/s^2 = 1000 kg * 9.8 m/s^2 * μ_rainy_day
μ_rainy_day = 0.94 m/s^2 / 9.8 m/s^2 ≈ 0.10
Therefore, the minimum coefficient of friction between the tires and the road that will allow cars to negotiate the turn without sliding off the road is approximately 0.10.