Inclination angle of a banked turn in a road for a maximum speed

[Andrei0408]

Homework Statement

In a real case, for a real radius curve and a real sliding
friction coefficient, find the inclination angle of a road for a maximum speed.(Ex. two-lane road, highway, racing circuit, railroad; tire-asphalt, meta-metal friction)

Relevant Equations

tan(theta)=(v^2)/r*g; μ=tg(alpha)

I know the solution is based on velocity and the sliding friction coefficient, and I believe I should put the condition Fcf smaller than Ff, but I just don't understand how to include μ in the solution, to find the angle. Even if you don't solve the problem, I just need to understand the concepts, please!

 

[Lnewqban]

Welcome, andrei0408!

What is that you don't understand specifically?
The vehicle naturally tends to keep going straight while the tires force it to follow a circular trajectory.
Friction force between the tires and the road is needed to achieve that change of direction.
The available friction force is certain percentage of the weight of the vehicle.
For different surfaces of the road, that percentage is called coefficient of friction (μ).

 

[Andrei0408]

Welcome, andrei0408!

What is that you don't understand specifically?
The vehicle naturally tends to keep going straight while the tires force it to follow a circular trajectory.
Friction force between the tires and the road is needed to achieve that change of direction.
The available friction force is certain percentage of the weight of the vehicle.
For different surfaces of the road, that percentage is called coefficient of friction (μ).

Well I need to find theta from the equation tan(theta)=(v^2)/r*g, but I also know that I need to include μ in order to solve for a real case.

 

[Lnewqban]

Please, see this:
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/carbank.html#c1

 

[berkeman]

Homework Statement:: In a real case, for a real radius curve and a real sliding
friction coefficient, find the inclination angle of a road for a maximum speed.

I must be missing something (is there another constraint?). As you go faster and faster the bank angle must rise to make your normal force support the car. Are there any other constraints? If not, then very fast speed is achieved with a maximum bank angle, it would seem.

Kind of like these guys:

 

[haruspex]

Homework Statement:: In a real case, for a real radius curve and a real sliding
friction coefficient, find the inclination angle of a road for a maximum speed.(Ex. two-lane road, highway, racing circuit, railroad; tire-asphalt, meta-metal friction)
Relevant Equations:: tan(theta)=(v^2)/r*g; μ=tg(alpha)

I know the solution is based on velocity and the sliding friction coefficient, and I believe I should put the condition Fcf smaller than Ff, but I just don't understand how to include μ in the solution, to find the angle. Even if you don't solve the problem, I just need to understand the concepts, please!

Since it specifies realistic situations, you should assume it is also a requirement to be able to go arbitrarily slowly on the same road without slipping down. That gives a max angle in terms of the coefficient.

 

[berkeman]

Since it specifies realistic situations, you should assume it is also a requirement to be able to go arbitrarily slowly on the same road without slipping down. That gives a max angle in terms of the coefficient.

Oh, interesting. That would definitely add a constraint.