Finding the bank of a racecar track

[Jrlinton]

Homework Statement

Forgive me as I am trying the recall this question by memory.
A racecar track is a perfect circle and 3000m long (I took this to be the distance traveled by cars, so the circumference) and after a day of time trials it is determined that the fastest possible time of completing one lap is 12 sec. What is the angle of the banking of the track?

Homework Equations

Vmax=(r*g*tan(theta))^.5

The Attempt at a Solution

First I found Vmax by dividing the 3000meters by 2 sec to get 250 m/s. Found r as 4712.4 m by dividing th circumference by 2pi. That makes the above equation 250=(4712.4*9.81*tantheta)^.5. Solve for theta=53.5 degrees.

 

[Jrlinton]

Okay correction the lap was completed in 32 seconds making Vmax 93.75 m/s and 3000/2pi is 477.47m. Solving for theta the angle being 62 degrees

 

[haruspex]

Okay correction the lap was completed in 32 seconds making Vmax 93.75 m/s and 3000/2pi is 477.47m. Solving for theta the angle being 62 degrees

I get the same answer. It seems huge, but this is assuming no friction(which raises the question of how the cars even get started). Are you sure there was no mention of friction in the original?

 

[Jrlinton]

Well the situation I am in is this was in a test that I was given today and the professor is allowing us a set time tomorrow to go back over for full credit with the freedom to use our memory and any outside sources in the meantime. I am almost certain that there was no coefficient of friction given. I understand your reasoning as friction must be present for a wheel to roll for a distance and that friction would play a factor in the banking, but is the friction not calculated indirectly by the given maximum velocity?

 

[haruspex]

is the friction not calculated indirectly by the given maximum velocity

No. With arbitrarily large friction coefficient, the speed is only limited by the risk of rolling sideways. No banking necessary.

 

[Jrlinton]

With the velocity regarding centripetal motion (as the track is a perfect circle) being in the direction of the tangent line, would the banking not be existing to keep the car from exiting through the outside of the track?

 

[haruspex]

With the velocity regarding centripetal motion (as the track is a perfect circle) being in the direction of the tangent line, would the banking not be existing to keep the car from exiting through the outside of the track?

Can you drive around a bend on a level, unbanked road?

 

[Jrlinton]

Right. So then how could that equation be correct at all? I mean it's one I got from the lectures.

 

[haruspex]

Right. So then how could that equation be correct at all? I mean it's one I got from the lectures.

The equation is correct for the no friction case. There is another equation you can write which gives the upper and lower limits of the speed for a given radius, bank angle and static friction coefficient.

 

[Jrlinton]

Do you have that equation on hand?

 

[haruspex]

Do you have that equation on hand?

It would be more beneficial for you to try to derive it yourself. The risk of just having a repertoire of equations is overlooking the contexts necessary for them to apply, as illustrated here.
Just consider the balance of forces, vertically and horizontally, in a vertical plane orthogonal to the velocity.