A banked circular highway curve is designed for traffic moving at 65 km/h. The radius of the curve is 207 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?
I can't figure this problem out. Any help (formulas, solutions, etc.) would be very much appreciated. Thanks!
To find the minimum coefficient of friction, we need to consider the forces acting on the car as it negotiates the curved highway. There are two main forces involved:
1. Centripetal force (F_c): This force keeps the car moving in a circular path and is provided by the friction between the tires and the road.
2. Gravitational force (F_g): This force acts vertically downward and is equal to the weight of the car.
The centripetal force can be calculated using the formula:
F_c = (m * v^2) / r
Where:
- F_c is the centripetal force
- m is the mass of the car
- v is the velocity of the car
- r is the radius of the curve
In this case, we need to calculate the minimum coefficient of friction, so we assume that the force of friction (F_f) is equal to the maximum value it can reach, which is the product of the coefficient of friction (μ) and the normal force (F_n).
F_f = μ * F_n
The normal force is equal to the gravitational force acting on the car:
F_n = F_g = m * g
Where:
- F_n is the normal force
- F_g is the gravitational force
- m is the mass of the car
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now we can substitute F_f and F_g into the equation for the centripetal force:
μ * F_n = (m * v^2) / r
μ * m * g = (m * v^2) / r
Now we can solve for the minimum coefficient of friction (μ):
μ = (v^2) / (r * g)
Let's substitute the given values into the equation:
v = 43 km/h = 43 * (1000 / 3600) m/s = 11.94 m/s
r = 207 m
g = 9.8 m/s^2
μ = (11.94^2) / (207 * 9.8) = 0.408
To solve this problem, we need to analyze the forces acting on the car as it negotiates the curved highway. The key force here is the friction force between the tires of the car and the road surface. This friction force provides the centripetal force required to keep the car moving in a circular path.
The centripetal force is given by the formula:
Fc = mv^2 / r
Where Fc is the centripetal force, m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
In this case, we need to find the minimum coefficient of friction (μ) that will prevent the car from sliding off the road. The friction force (Ff) can be calculated using the formula:
Ff = μN
Where Ff is the friction force and N is the normal force. On a level road, the normal force is equal to the weight of the car (mg), where g is the acceleration due to gravity.
Now, let's break down the problem into steps:
Step 1: Convert the given speeds from km/h to m/s:
- Traffic speed on the highway = 43 km/h
- Curve speed design = 65 km/h
Speed on the curve (v): v = 43 km/h = (43 * 1000) / (60 * 60) m/s ≈ 11.94 m/s
Step 2: Calculate the centripetal force (Fc):
Fc = mv^2 / r
Step 3: Calculate the normal force (N):
N = mg
Step 4: Calculate the friction force (Ff):
Ff = μN
Step 5: Set up the inequality to find the minimum coefficient of friction:
μ ≥ Fc / N
Now, let's proceed to calculate the values.
Step 1 (already done): v = 11.94 m/s
Step 2:
Given the radius of the curve (r) = 207 m and the speed (v) = 11.94 m/s:
Fc = mv^2 / r
Step 3:
The weight of the car (N) = mg, where m is the mass of the car and g is the acceleration due to gravity (9.8 m/s^2). The mass is not provided in the problem, so we cannot calculate it directly. However, we can see that the mass cancels out when we divide Fc by N in Step 5. This means we do not need the mass value to find the minimum coefficient of friction.
Step 4:
Ff = μN
Step 5:
To find the minimum coefficient of friction, we need to use the inequality:
μ ≥ Fc / N
Plugging in the values calculated in previous steps, we get the equation:
μ ≥ (mv^2 / r) / N
Since the mass (m) cancels out, we can simplify the equation to:
μ ≥ v^2 / (rg)
Now, plug in the given values and calculate the minimum coefficient of friction:
μ ≥ (11.94^2) / (207 * 9.8)
μ ≥ 0.786
Therefore, the minimum coefficient of friction between the car's tires and the road should be at least 0.786 to allow the cars to negotiate the turn without sliding off the road.