How Do You Calculate the Minimum Coefficient of Friction for a Banked Curve?

[nickmai123]

"A banked circular highway curve is designed for traffic moving at 70 km/h. The radius of the curve is 186 m. Traffic is moving along the highway at 35 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?"


$$F_{c}=\frac{mv^{2}}{r}$$

$$v_{unbanked}=\sqrt{ru_{k}g}$$

$$v_{banked}=\sqrt{rgtan\theta}$$


I'm lost. I don't know where to start with this problem. I know how to work the typical banked and unbanked curves, but I don't get how to draw the free-body diagram here.

Thanks.

 

[jgens]

Well, a good place to start this problem is finding an expression for the banking angle (the case without friction). Draw the freebody diagram for this simplified problem and obtain an expression for the banking angle.

Once you've done that, draw the freebody diagram including friction. Note that the only new force acting in this picture is the frictional force and that it acts parallel to the banked surface.

 

[tomcue]

As a scientist, it is important to approach problems with a systematic and analytical mindset. In this scenario, we are dealing with a banked curve with friction, which means we need to consider both the effects of the banked angle and the friction between the tires and the road. Let's break down the problem and see how we can solve it step by step.

First, we need to understand that the banked angle, or the angle at which the curve is inclined, is designed for a specific speed of 70 km/h. This means that at this speed, the centripetal force (F_c) generated by the curve is equal to the force of gravity (mg) pulling the car down the banked curve. This can be expressed as F_c = mg.

Next, we need to consider the forces acting on the car as it moves along the curve. These forces include the normal force (N) exerted by the road on the car, the force of gravity (mg), and the frictional force (F_f) between the tires and the road. Since the car is moving at a constant speed of 35 km/h, we can assume that the net force acting on the car is zero, meaning the sum of all forces in the horizontal direction is equal to zero.

Now, let's draw a free-body diagram to better visualize these forces. The normal force (N) acts perpendicular to the road, while the force of gravity (mg) acts straight down towards the center of the curve. The frictional force (F_f) acts in the opposite direction of the car's motion, as it is trying to prevent the car from sliding off the road. Keep in mind that the angle of the banked curve (θ) is also acting on the car, as it is trying to push the car towards the center of the curve.

Using the equations provided in the problem, we can set up a system of equations to solve for the minimum coefficient of friction (u_k) between the tires and the road. Since we know the speed (v) and radius (r) of the curve, we can plug these values into the equations and solve for u_k.

Once we have the value of u_k, we can compare it to known coefficients of friction for different road conditions (e.g. dry road, wet road, etc.) to determine if the cars can safely negotiate the curve without sliding off the road.

In summary, solving this problem requires a thorough

 

[nlightner]

I understand your confusion and I am here to guide you through the problem. Let's start by breaking down the given information. We have a banked circular highway curve designed for traffic moving at 70 km/h. This means that the curve is sloped at an angle (θ) to allow the cars to safely navigate the turn at that speed. The radius of the curve is 186 m, and the traffic is moving at a slower speed of 35 km/h on a rainy day. We are asked to find the minimum coefficient of friction between the tires and the road that will prevent the cars from sliding off the road.

To solve this problem, we need to consider the forces acting on the car as it navigates the banked curve. These forces include the centripetal force (Fc), the normal force (N), and the frictional force (Ff). The centripetal force is responsible for keeping the car moving in a circular motion, while the normal force is perpendicular to the surface of the road and balances the weight of the car. The frictional force is what we are interested in as it determines the amount of friction between the tires and the road.

To find the minimum coefficient of friction, we can use the equations mentioned in the problem statement. The first equation, Fc = mv^2/r, represents the centripetal force and is equal to the mass (m) of the car multiplied by its velocity (v) squared and divided by the radius (r) of the curve. The second equation, vunbanked = √(ruk), represents the velocity of an unbanked curve, where u is the coefficient of friction and k is the acceleration due to gravity. The third equation, vbanked = √(rgtanθ), represents the velocity of a banked curve, where g is the acceleration due to gravity and θ is the angle of the banked curve.

Now, to solve for the minimum coefficient of friction, we need to set the two velocities (vunbanked and vbanked) equal to each other. This is because the car is not sliding on the road, meaning that the two velocities are equal. We can then substitute the values given in the problem and solve for u.

35 km/h = √(9.8 m/s^2 * 186 m * u)
u = 0.35

Therefore, the minimum coefficient of friction between the