As a civil engineering intern during one of your summer in college, you are asked to design a curved section of roadway that meets the following conditions: when ice is on the road, and the ceefficient of static friction between the road and rubber is 0.080, a car at rest must not slide into the ditch and a car traveling less than 60km/h must not skid to the outside of the curve.Neglect the effects of air drag and rolling friction.What is the minimum radius of curvature of the curve and at what angle should the road be banked?
To determine the minimum radius of curvature and the angle at which the road should be banked, we need to analyze the forces acting on the car as it moves along the curved roadway.
1. Consider the car at rest on the icy road surface. In this case, we need to ensure that the car does not slide into the ditch. The limiting friction force that prevents sliding can be calculated using the formula:
F_friction = μ_s * N
where F_friction is the friction force, μ_s is the coefficient of static friction, and N is the normal force acting on the car.
The normal force can be decomposed into two components: the vertical component (N_vert) and the horizontal component (N_hor). At rest on a level road, N_vert equals the weight of the car, which can be calculated as the product of the car's mass (m) and the acceleration due to gravity (g).
N_vert = m * g
Therefore, N_hor = N_vert * sin(θ) where θ is the banking angle.
The maximum friction force required to prevent sliding can be expressed as:
F_max = μ_s * N_hor = μ_s * N_vert * sin(θ)
To avoid sliding into the ditch, the force required to keep the car on the road (centripetal force) must not exceed the maximum friction force:
F_centrifugal ≤ F_max
The centripetal force is given by:
F_centrifugal = m * v^2 / r
where m is the mass of the car, v is the velocity, and r is the radius of curvature.
2. Now, consider a car traveling at less than 60 km/h (which is equivalent to 16.7 m/s). In this case, we need to ensure that the car does not skid to the outside of the curve. Skidding occurs when the frictional force is not sufficient to provide the necessary centripetal force to keep the car on the curved path.
The maximum friction force required to prevent skidding can be expressed as:
F_max = μ_s * N_vert
The centripetal force required to keep the car on the road is the same as before:
F_centrifugal = m * v^2 / r
To prevent skidding, the friction force must be greater than or equal to the centripetal force:
F_max ≥ F_centrifugal
Now, we can substitute the expressions for F_max and F_centrifugal and solve for r:
μ_s * N_vert ≥ m * v^2 / r
Combining expressions for N_vert and N_hor:
μ_s * m * g ≥ m * v^2 / r * sin(θ)
Simplifying:
r ≥ (μ_s * g * v^2) / (g * sin(θ))
r ≥ (μ_s * v^2) / sin(θ)
This is the minimum radius of curvature needed for a given speed and banking angle.
To find the angle at which the road should be banked, we can rearrange the formula above and solve for θ:
θ = arcsin((μ_s * v^2) / r)
Plug in the values for the coefficient of static friction (0.080) and the velocity (60 km/h or 16.7 m/s), then substitute the minimum radius of curvature to find the angle of banking (θ).
To find the minimum radius of curvature and the required banking angle for the curved section of roadway, we can apply the concepts of centripetal force and frictional force. Let's break down the problem step-by-step:
Step 1: Identify the forces acting on the car:
When a car is moving along a curved road, two primary forces come into play: the frictional force and the gravitational force.
- Frictional force: The static friction between the rubber tires and the icy road helps provide the necessary centripetal force to keep the car on the curve.
- Gravitational force: This force acts vertically downwards and needs to be taken into account while determining the banking angle.
Step 2: Determine the critical conditions for design:
To design the road safely, we need to consider two critical conditions:
1. Car at rest: The car should not slide into the ditch when it is at rest.
2. Car traveling below 60 km/h: The car should not skid to the outside of the curve when it is traveling at speeds less than 60 km/h.
Step 3: Analyze the forces acting on the car at rest:
When the car is at rest, the frictional force should be sufficient to prevent it from sliding into the ditch. To calculate the minimum coefficient of static friction, we can relate the gravitational force to the frictional force using the following equation:
F_friction = F_gravity
The frictional force can be calculated as:
F_friction = μ * F_normal
where:
μ = coefficient of friction = 0.080 (given)
F_normal = normal force = mg, where m is the mass of the car and g is the acceleration due to gravity.
Step 4: Determine the normal force:
To calculate the normal force, we need to consider the forces acting vertically on the car.
F_normal = F_gravity = mg
Step 5: Calculate the minimum speed required to prevent skidding:
When the car is traveling below 60 km/h, it should not skid to the outside of the curve. This means that the frictional force should be sufficient to provide the necessary centripetal force to keep the car on the curve.
The centripetal force is given by:
F_centr = m * v^2 / r
where:
m = mass of the car
v = velocity of the car
r = radius of curvature
The frictional force can be calculated as:
F_friction = μ * F_normal
Step 6: Determine the angle of banking:
To determine the angle of banking, we need to consider the horizontal forces acting on the car. The angle of banking, θ, is given by the equation:
tan(θ) = v^2 / (g * r)
where:
v = velocity of the car
g = acceleration due to gravity
r = radius of curvature
Step 7: Solve for the minimum radius of curvature and angle of banking:
By combining the equations for the two critical conditions, we can solve for the minimum radius of curvature and the angle of banking simultaneously.
Since we are neglecting the effects of air drag and rolling friction, we can assume that there is no external force acting horizontally on the car. Hence, the net horizontal force is zero.
F_friction = m * g * tan(θ)
Combining this with the equation for F_friction from Step 5, we get:
μ * F_normal = m * g * tan(θ)
μ * m * g = m * g * tan(θ)
μ = tan(θ)
Plug in the given coefficient of static friction (μ = 0.080) into the equation:
0.080 = tan(θ)
Now, solve for the angle of banking, θ.
Once you have the angle of banking, you can rearrange the equation for the centripetal force from Step 5 to solve for the minimum radius of curvature, r:
r = m * v^2 / (g * F_friction)
Substitute the given parameters and the calculated value of F_friction to find the minimum radius of curvature.
That's it! By following these steps, you can determine the minimum radius of curvature and the angle of banking required for the curved section of roadway, given the specified conditions and coefficients.
Well, my first thought is that of road engineers: Banked curves beg speeders to speed. One cannot on a public highway encourage racing.
Now, the question.
Friction down the incline= mg*.8*cosTheta
so gravity down the incline cant be greater than that.
(1) mgSinTheta<mg*.8*cosTheta
or TanTheta<.8
Now at 60km/hr (16.7m/s) centripetal force will have two components: up the plane (mv^2/r * CosTheta) and normal to the plane (mv^2/r*sinTheta). We still have the up the plane and normal components of the weight of the car.
So, the force up the plane < retarding friction force.
mv^2/r*CosTheta -mgSinTheta < .8(mgCosTheta+mv^2/r * SinTheta)
Divide both sides by cosTheta
mv^2/r -mg TanTheta<.8mg+.8mv^2/r tanTheta
solve for tan Theta in (1)put that into the equation and solve for r.