# Banked Curves

### Overview

When was the last time you watched an auto race on TV? It’s no wonder it’s such a popular sport, with cars going over 200 mph around the course. If you tried to do that on the highway, your car would probably go sliding off the road at the first turn. (That’s if it could even go that fast!)

One thing that really helps the drivers stay on the course is that the curves are banked. This means that they slope inward, towards the direction of the turn. In this way, the force of gravity tends to pull the car towards the inside of the turn, reducing the amount of friction needed by the tires to keep the car from sliding off the road.

Just how much do they bank those turns? Well, at the Texas Motor Speedway, the site of several car races, the curves are banked 28 degrees. That’s a lot!

Normal roads are banked too, but not that much. Next time you’re on the highway and you go around a corner, notice whether or not it is banked. If it’s a fairly new road, there’s a good chance that it is. This makes the road much safer to drive on and also much more comfortable. You won’t feel that sideways force as much on a banked curve.

We can use the vortex in Chaos to illustrate and understand banked curves.

#### Setup

Set up the vortex as shown below.

### Activity

#### Stage 1

Roll a ball down the track and into the vortex. Notice how the slope of the track causes the ball to roll around in a circle. This slope allows gravity to pull on the ball, but always in a direction towards the center of the vortex. Because the force is sideways to the ball’s motion, the ball is changing direction, but it’s speed remains the same. (Actually, the ball slowly slows down due to losses from friction. This causes it to move closer to the center.)

#### Stage 2

This is a good time to point out the difference between speed and velocity. To specify and objects velocity, we need both its speed and direction. In this case the ball’s velocity is constantly changing because it’s direction is always changing. However, it’s speed is not changing. (At least not very much.)

#### Stage 3: A Quantitative Understanding of a Racetrack

Using the laws of physics we can conclude how fast the ball is rolling if we know the angle of the incline of the vortex and radius of the curved path the ball is following. The formula is:

V = Sqrt(r*g*sin q)

Where V is the speed, r is the radius of the curve (in meters), g is the gravitational acceleration of 9.8 m/s², and q is the slope of the vortex.

Use this formula to estimate how fast the ball is rolling when it is three inches from the center of the vortex. You need to estimate the angle of the vortex at this distance.

Hints:

The angle is about 30 degrees – just use this value

Remember to convert from inches to meters: 1 inch = 2.54 cm

Record what you get for a speed here

Calculated Speed = ________

Note: Using the value above, the answer is 50 cm/s.

Does this seem reasonable?

Let’s check this with a measurement. Carefully mark a spot on the vortex two inches from the center. (This is s inches horizontally.) Roll the ball and mark and use a stopwatch to measure the time it takes to go around the vortex when the radius is two inches. It might be good to measure the time it takes to go around 3 times and then divide that number by three. This will help you get a more accurate result.

We want a value for speed, which is distance per time. The distance the ball travels is the circumference, which is 2*Pi*r. For r = 2 inches, this is 32 cm. The time is whatever you measured it to be.

So the speed is : V= 32/time = _______ cm/s

How well does this agree with your calculation above?

### Quiz Yourself

#### Stage 1

Suppose you were to conduct the same measurement as above, but using a heavy gold ball instead of the yellow you are using now. How would this affect how long it takes to go into the hole?

[It wouldn’t change them, because the result doesn’t depend on the mass of the ball. The answer only depends on the radius and angle of the vortex.]

#### Stage 2

How would the results change if you used a ball that was twice as large? Twice as small?

[Again, the results wouldn’t change]

#### Stage 3

A race car drive is driving on the track at the Texas Motor Speedway. The only problem is that the track is completely covered with ice, meaning that he will not have any traction in the turn. If the radius of the turn is 200 meters, how fast should he go to not slide off the track? How fast is this in miles per hour?

Solution:

a) V=Sqrt(r*g*sin q)

V=Sqrt(200*9.8*sin(28))

V=30.3 m/s

b) convert 30.3 m/s to miles per hour

30.3 m/sec

60 sec = 1 min

60 min = 1 hour

100 cm = 1 m

1 in = 2.54 cm

1 ft. = 12 in.

1 mile = 5280 ft.

67.8 mph