Analysing a Circular Highway Curve: Angle, Friction & Speed

[chiurox]

Homework Statement

A circular highway curve with a radius of 200m is banked at an angle such that a car traveling 45km/h can just make it around the curve if the highway surface is frictionless.
a. what is the angle between the highway surface and the horizontal?
b. if a car travels at 40km/h around the curve, what is the centripetal force acting on the car?
c. What is the minimum value of the coefficient of friction between the tires and the highway surface necessary to prevent the car in (b) from skidding?
d. If the angle of the highway curve is as in (a), but the radius of the curve is increased to 300m, what is the speed a car must be going in order to negotiate a curve without skidding?

Homework Equations

tan(theta) = v^2/gr

The Attempt at a Solution

45km/h would be 12.5m/s
a. tan(theta) = (12.5m/s)^2 / 9.8*200
theta = 4.6degrees
b. I know that F_n sin(theta) = mv^2/r
but I'm confused how to solve this one.
c. I'll leave this blank since I haven't solved (b)
d. sqrt{tan(4.6)(9.8)(300)} = v = 15.4m/s correct?

 

[chiurox]

For part C, I have F_f = u_s F_n
v^2 = u_s gr
1600 = u_s(9.8)(200)
u_s = 0.816 right?

For part D, I have sqrt(tan(4.5)(9.8)(300)) = v = 15.2 m/s is that right?

 

[Moetasim]

I would like to provide a comprehensive response to the content provided. Firstly, let's start with the given information. We have a circular highway curve with a radius of 200m, banked at an angle that allows a car traveling at 45km/h to make it around the curve without any friction on the highway surface. From this, we can determine the angle between the highway surface and the horizontal to be 4.6 degrees, as calculated in the attempt at a solution.

Moving on to the next part of the problem, we are asked to find the centripetal force acting on a car traveling at 40km/h around the curve. To solve this, we need to use the formula Fc = mv^2/r, where Fc is the centripetal force, m is the mass of the car, v is the velocity, and r is the radius of the curve. However, we are not given the mass of the car, so we cannot solve this part of the problem.

Next, we are asked to find the minimum value of the coefficient of friction between the tires and the highway surface necessary to prevent the car from skidding at a speed of 40km/h. To solve this, we need to use the formula Ff = μmg, where Ff is the frictional force, μ is the coefficient of friction, m is the mass of the car, and g is the acceleration due to gravity. Again, we are not given the mass of the car, so we cannot solve this part of the problem.

Finally, we are asked to find the speed a car must be going in order to negotiate a curve without skidding, given that the angle of the highway curve is 4.6 degrees and the radius is increased to 300m. To solve this, we can use the formula v = √(rgtanθ), where v is the required speed, r is the radius of the curve, g is the acceleration due to gravity, and θ is the angle between the highway surface and the horizontal. Plugging in the given values, we get v = 15.4m/s, as calculated in the attempt at a solution.

In conclusion, as a scientist, I would like to point out that in order to solve these problems accurately, we need to have all the necessary information, such as the mass of the car, in order to apply the relevant formulas. Without this information,