The goal of this project was to find what shape a roller coaster has to have to provide a constant centripetal force on the rider.
Note: this file is best viewed in light-mode.
For an object to move in a circle, it must have an inward force applied on it.
This is called a "center-seeking" or centripetal force.
For an object of mass m moving at a velocity v in a circular path of radius r, the force F required to maintain this motion is given by .
Before a roller coaster goes down, it must first go up.
A roller coaster of mass m at a height H will have a potential energy U of where g is the gravitational acceleration on earth (about 9.801 m/s^2).
Then, let the roller coaster fall down.
If we write the height of the roller coaster as a function of time h(t), the new potential energy is .
However, by conservation of energy, the kinetic energy K will be
.
Recalling that our initial potential energy is
, and the formula for the kinetic energy of our roller coaster moving at velocity v is
, we get
.
Define time is 0 to be when the roller coaster is at the lowest point on the track.
Here, , so
.
Let the radius of curvature at this lowest point on the track be R.
Then the centripetal force at time 0 is
.
Since
, we can write
.
Let the radius of curvature be r(t) other places along the loop.
Then, the centripetal force at other places along the loop is
.
To maintain a constant centripetal force, we require
, so we get
.
A little rearranging gives
.
Therefore, we have the radius of curvature as a function of height. This restriction is all we need to construct a loop given the initial H and R conditions.
The Loop.py program creates the loop by starting at the bottom (when radius of curvature is R) and ensure that at each step, .
Here is a sample output where H = 100 meters and R = 50 meters:
Some interesting combinations of (H, R) to try are (100, 0.5), (100, 10), (100, 100), and (100, 200). You may notice that these combinations all produce outputs that resemble roller-coaster loops you've seen in real life!
In addition to Loop.py, there is also Loop_Height.py. This program creates loops H = 100 meters and radii of curvatures ranging from 1 meter to R_max meters, which is currently set to 200 meters (the increment is 1 meter). A file Loop_Heights.txt is created to store the maximum height each loop reaches (which is not H because if the height reaches H, then there cannot be any more kinetic energy, meaning velocity is 0, so the centripetal force is also 0; however, that cannot be as the roller coaster started with some non-zero centripetal force). Another file Loop_Coordinates.txt is used to store the (x, y) distances from where the roller coaster hits the lowest point to when it hits its heighest point. This can be considered the "coordinate" of the peak of the loop if you set the origin to be where the roller coaster hits the lowest point.
Finally, it plots three graphs in order: the maximum loop height given the radius of curvature, the x-coordinate of the peak of the loop given the radius of curvature, and the x vs y coordinates of the peak of the loop for loops with different inital radii of curvature. Here is a sample output given R_max = 200 meters:
You may notice that the graph of the initial radius of curvature vs maximum loop height follows a logistic relation. This is an interesting note that bears further investigation.