Physics of Daytona

[MolaKule]

The Daytona Speedway features four-story, 31 degree (theta) banked curves with a maximum radius (r) of 316 meters.

Two forces act on the car, gravity, g (a downward force), and the Normal Force, N (a vertical force).

We will neglect tire friction since the COF between the pavement and tires is close to 1. We will also neglect aerodynamic drag.

Find the Centripetal acceleration necessary to keep the car from sliding up or down the track.

m is the mass of the car.
m.a = Summation of all forces = N + m.g
A Period “.” means multiplication.
N.cos(theta) – m.g = 0.
N = m.g/cos(theta)

Fc is centripetal Force.

The centripetal force Fc = n.sine(theta) = m.g.sine(theta)/cos(theta) = m.g.tan(theta)

Ac is centrepital acceleration..

Ac = Fc/m = m.g.tan(theta)/m = g.tan(theta) = (9.80 m/s^2).(tan(31degrees)) = 5.89 m/s^2. This is the centripetal acceleration necessary to keep the race car from sliding up or down the track.

Since Ac also equals v^2/r, the speed of the race car will have to be v = sqrt(r.Ac) = 43.1 m/s or 96.41 miles/hour. Any speed faster than 96.41 mph will cause the race car to drift up the bank, while a speed less than 96.41 mph will cause the race car to drift down the bank.

Question: What happened to the mass, m, of the car in determining the centripetal acceleration and speed, and what does this mean?

 

[gregk24]

Way over my head.

 

[abycat]

had to replace an oven today cause I couldn't fix it. Actually I got fed up and said I give up. Just like I gave up trying to figure out mola's riddles. ha ha you crazy guy you.

 

[Nyquist]

Hint: Sum of all forces (all vectors) must equal zero assuming constant speed.

 

[Phishin]

Gravity has the same force regardless of mass.

 

[OneEyeJack]

43.10989 meters per second or about 96 mph should do the trick.

 

[EdwardC]

Originally Posted By: Phishin
Gravity has the same force regardless of mass.


Yep.

That post would be way easier with the LaTeX message board plugin!

 

[Shannow]

Your third sentence says to ignore tyre friction as u is close to 1, while your analysis is based on u=0, the car having no forces between the track and tyre other than the normal force acting perpendicular to the track surface, and up through the CoG of the car...

Given that the forces due to the centripetal acceleration, and the gravitational force are proportional to mass, the masses cancel each other out as soon as you go back to calculate acceleration...

 

[A_Harman]

Yep, fun with physics, especially when applied to automobiles.

Race tracks I'd like to see:
1. A simple oval with 90 degree banking at each end and very carefully constructed spiral easements. This would basically be an infinitely long straightaway. Top end only limited by horsepower and bravery. Mercedes has a test track with 90 degree banking which they call "the Wall of Death".
2. A track with an inverted straightaway. Picture a half-pipe that connects tangent to an inverted flat surface so that cars that have higher downforce than weight could literally "drive across the ceiling". The race car engineers better do their homework on lube, coolant, and fuel system design so that the car keeps running when it turns upside-down.
3. A track with a loop-de-loop in it. Come on, everybody had one in their Hot Wheels sets when they were kids. Wouldn't we all like to see one in full scale?

 

[Garak]

Originally Posted By: MolaKule
Ac = Fc/m = m.g.tan(theta)/m = g.tan(theta) = (9.80 m/s^2).(tan(31degrees)) = 5.89 m/s^2.

The mass cancels itself out here.

 

[dailydriver]

Originally Posted By: A_Harman
Yep, fun with physics, especially when applied to automobiles.

Race tracks I'd like to see:
1. A simple oval with 90 degree banking at each end and very carefully constructed spiral easements. This would basically be an infinitely long straightaway. Top end only limited by horsepower and bravery. Mercedes has a test track with 90 degree banking which they call "the Wall of Death".


Thanks, now I have Richard Thompson's "Wall Of Death" playing in my head on repeat.

 

[MolaKule]

The mass does cancel out, because mass does not come into play.

The important items to consider here are:

1.) the centripetal force, n.sin(theta), points inward the center of the radius and accelerates the vehicle towards the center.

2.) The bank is angled, otherwise the car would fly off at a tangent, i.e, it would attempt to continue in a straight line if not for the centripital acceleration inward toward the center,

3.) The coefficient of friction between the tires and pavement is almost equal to 1 in this case, which is why we neglected it.

TO determine the MINIMUM coefficient of friction Us necessary for this speed and radius of curcvature, Us = v^2.cos(theta)/g.r.

Putting in the values we get Us = 0.51.

 

[MolaKule]

Originally Posted By: EdwardC
Originally Posted By: Phishin
Gravity has the same force regardless of mass.


Yep.

That post would be way easier with the LaTeX message board plugin!


True, but you use what you have.

 

[MolaKule]

Quote:
3. A track with a loop-de-loop in it. Come on, everybody had one in their Hot Wheels sets when they were kids. Wouldn't we all like to see one in full scale?


Well, I was one of those poverty stricken children who didn't have one. Well, they hadn't quite invented them yet.

But I made sure my grandsons had them by golly.

I'll design it, you build it and find some brave drivers.

 

[A_Harman]

Originally Posted By: MolaKule
Quote:
3. A track with a loop-de-loop in it. Come on, everybody had one in their Hot Wheels sets when they were kids. Wouldn't we all like to see one in full scale?


Well, I was one of those poverty stricken children who didn't have one. Well, they hadn't quite invented them yet.

But I made sure my grandsons had them by golly.

I'll design it, you build it and find some brave drivers.

That's the ultimate test of how much an engineer believes in the thing he's designing; is he willing to be the first to use it? There used to be a rule in the airplane business: the Chief Designer of an airplane goes along for the first flight. You tend to be very conservative and doublecheck your math if your life is on the line.

 

[Garak]

Originally Posted By: MolaKule
The mass does cancel out, because mass does not come into play.

Yes, and we can simplify even further to demonstrate that, just using a single axis. Someone standing on the ground motionless has their weight given as F(g)=mg, with special notice that g will be a negative number, pointing down the coordinate system. The normal force will be F=-mg. Whatever is standing on the surface won't move up or down because the gravitational force and the normal force will balance each other.

When dealing with vectors, even in one dimension, this is an example why I don't like to get rid of the cosine. If you use your angles in your coordinate system rigorously, following the rules of the unit circle, the negatives and positives will take care of themselves every time. The cosine of 0 is 1, so that won't matter. For the force in the opposite direction, pi radians is upside down, and the cosine of that is -1, taking care of your signs automatically.

 

[Shannow]

Originally Posted By: MolaKule
3.) The coefficient of friction between the tires and pavement is almost equal to 1 in this case, which is why we neglected it.


If you are using friction, and it's nearly equal to 1, you can't ignore it...it's the same as turning 1g on the skid-pad...no bank needed to stop it flying off.

In reverse, if U=1, the car can stop on the bank and not slide down to the infield...otherwise flags at Daytona would wreak more carnage than what caused the flag.

 

[MolaKule]

Now that we have all the math completed, one can use a Us of 0.9 and calculate the real world safe MAXIMUM speed in m/s or mph.

Multiply m/s by 2.24 to get the mph.

 

[MolaKule]

Originally Posted By: A_Harman
Originally Posted By: MolaKule
Quote:
3. A track with a loop-de-loop in it. Come on, everybody had one in their Hot Wheels sets when they were kids. Wouldn't we all like to see one in full scale?


Well, I was one of those poverty stricken children who didn't have one. Well, they hadn't quite invented them yet.

But I made sure my grandsons had them by golly.

I'll design it, you build it and find some brave drivers.

That's the ultimate test of how much an engineer believes in the thing he's designing; is he willing to be the first to use it? There used to be a rule in the airplane business: the Chief Designer of an airplane goes along for the first flight. You tend to be very conservative and doublecheck your math if your life is on the line.


I'll design the track, you design the race car chassis and engine, we'll compare our design equations, double check them, and then we will BOTH go for that thrill of the first test ride.

But I will still insist on a professional driver.

[BTW, when I was doing Aerospace Engineering, I WAS on the first test flights for the 737NG, some modified military aircraft. and business jets.]

 

[badtlc]

Originally Posted By: Phishin
Gravity has the same force regardless of mass.


Gravity has the same acceleration, not force. Force = mass x accleration. more mass equals more force.