A banked circular highway curve is designed for traffic moving at 60 km/h. The radius of the curve is 207 m. Traffic is moving along the highway at 42 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to take the turn without sliding off the road? (Assume the cars do not have negative lift.)
To determine the minimum coefficient of friction between the tires and road that will prevent cars from sliding off the road on a banked circular highway curve, we need to consider the forces acting on the car.
1. First, let's find the angle of banking (θ) for the curve. We can use the formula:
θ = arctan(v^2 / (g * r))
where v is the velocity and r is the radius of the curve, and g is the acceleration due to gravity.
Converting the given velocities to meters per second (m/s) and the radius to meters (m):
v = 42 km/h = (42 * 1000) / (60 * 60) m/s ≈ 11.67 m/s
r = 207 m
g = 9.8 m/s^2
Now, we can calculate the angle of banking:
θ = arctan((11.67^2) / (9.8 * 207))
2. Next, let's calculate the net force acting on the car along the vertical axis. There are two main forces that contribute to this:
a. Weight (W) acting vertically downwards: W = m * g, where m is the mass of the car.
b. Centripetal force (Fc) acting towards the center of the curve: Fc = m * v^2 / r
The net force along the vertical axis is given by:
Fnet = Fc - W*cos(θ)
where θ is the angle of banking.
3. To prevent sliding, the net force in the vertical direction should be equal to the frictional force, which is given by:
Ffriction = μ * W*sin(θ)
where μ is the coefficient of friction.
Since the car is not sliding, Fnet = Ffriction.
4. Now, substitute relevant values and solve the equation for μ:
m * v^2 / r - m * g*cos(θ) = μ * m * g*sin(θ)
Simplifying the equation:
μ = (v^2 - g * r * cos(θ)) / (g * sin(θ))
Plug in the given values and the angle of banking you calculated.
μ = (11.67^2 - 9.8 * 207 * cos(θ)) / (9.8 * sin(θ))
Finally, evaluate the expression to find the minimum coefficient of friction required for the cars to take the turn without sliding off the road.