The design of a new road includes a straight stretch that is horizontal and flat but that suddenly dips down a steep hill at 25°. The transition should be rounded with what minimum radius so that cars traveling 97 km/h will not leave the road?
To determine the minimum radius of the transition, we need to consider the forces acting on the car as it goes through the dip. Specifically, we need to ensure that the centripetal force being provided by the curve is enough to keep the car on the road.
Here are the steps to calculate the minimum radius:
1. Identify the forces acting on the car in the dip:
- Weight force acting vertically downward (mg)
- Normal force acting perpendicular to the road surface (N)
- Friction force acting horizontally (f)
- Centripetal force acting towards the center of the curve (Fc)
2. Calculate the centripetal force (Fc):
The centripetal force is provided by the friction force. It can be calculated using the equation: Fc = m * v^2 / r, where m is the mass of the car, v is its velocity, and r is the radius of the curve. Rearranging this equation, we have: f = m * v^2 / r
3. Calculate the maximum possible friction force (f):
The maximum friction force that can be provided without the car skidding can be calculated using the equation: f = μ * N, where μ is the coefficient of friction between the tires and the road surface, and N is the normal force acting on the car.
4. Calculate the normal force (N):
The normal force can be calculated using the equation: N = m * g, where m is the mass of the car and g is the acceleration due to gravity.
5. Substitute the value of the normal force (N) in the equation for maximum friction force (f):
f = μ * N = μ * m * g
6. Substitute the value of the maximum friction force (f) in the equation for the centripetal force (Fc):
f = m * v^2 / r
7. Equate the two equations for the friction force (f) to find the minimum radius (r):
μ * m * g = m * v^2 / r
8. Rearrange the equation to solve for the radius (r):
r = v^2 / (μ * g)
Now, let's substitute the given values:
Velocity (v) = 97 km/h = 97 * (1000/3600) m/s ≈ 26.94 m/s
Coefficient of friction (μ) = Since it is not given, we will assume a realistic value of 0.7
Acceleration due to gravity (g) ≈ 9.8 m/s^2
Substituting these values into the equation, we get:
r ≈ (26.94^2) / (0.7 * 9.8) ≈ 101.17 meters
Therefore, the minimum radius of the transition should be approximately 101.17 meters to ensure that cars traveling at 97 km/h will not leave the road.
To calculate the minimum radius required for the transition, we need to consider the centripetal force acting on the car as it goes through the curve. The maximum force the car can withstand without leaving the road is equal to the weight of the car in this case. We can use the following formula to calculate the centripetal force:
Fc = mv² / 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.
First, we need to convert the velocity from km/h to m/s:
97 km/h = 97 x (1000 m / 3600 s) = 26.94 m/s
Next, we need to calculate the weight of the car. Let's assume a typical mass for a car, such as 1500 kg, and use the acceleration due to gravity (9.8 m/s²) to find the weight:
Weight = mass x acceleration due to gravity
= 1500 kg x 9.8 m/s²
= 14700 N
Now we can rearrange the formula to solve for the radius:
r = mv² / Fc
Plugging in the values:
r = (1500 kg x (26.94 m/s)²) / 14700 N
= (1500 kg x 726.36 m²/s²) / 14700 N
≈ 73.52 m
Therefore, the minimum radius required for the transition should be approximately 73.52 meters to ensure that cars traveling at 97 km/h will not leave the road.