Mechanics - "loop the loop"

Mechanics 2 “loop the loop”

Task

What is the minimum release height needed to “loop the loop.” Investigate and validate.

To do this I will create a theoretical model to calculate the minimum height required for the roller coaster to just reach the top of the loop then look at the height required to fully complete a loop.

I will validate my theoretical model using a track.

Assumptions

So I can make a relatively simple model I will have to make some assumptions:

There is no friction because if there was there would be some work done against it and would mean I could not use the conservation of energy. Roller coasters run on wheels against smooth metal which would not have much friction.
Air resistance is negligible as most roller coasters are designed quite aerodynamically and have a large mass which would not be affected by a significant amount.
No energy is lost e.g. sound and heat which would be a result of friction.
There are no resistive forces
The loop is an exact circle to make calculations easier and allows me use what I know about circular motion. My model loop starts and finished at the same place which could not happen in reality because the roller coaster could either not get on the loop or would do continuous loops.
The value for gravity is constant. In reality it changes depending on the distance you are from the centre of the earth but the roller coaster does not have a big enough height difference between its highest and lowest points. This is the formula for calculating as value for gravity:

g = G m

r²

For example the value of gravity at the top of the roller coaster Kraken at Seaworld Orlando which is 45m above the earths surface is – 9.7999 this is almost the same as the value of gravity on the earths surface so will not make any significant difference to my model. I will use 9.8 ms-2 in my calculations.

The roller coaster is a particle and not a long multi car carriage. This is to make calculations easier.
Natural variables like the wind will be ignored.
In my model I will use a straight ramp (see diagram below) leading to the loop rather than a curved section that a real life roller coaster would have. This is to make calculations easier.
In my model I will assume that the roller coaster starts from rest.

Model

Symbols I will use in my calculations:

r = radius of the loop
k = constant co-efficient of r for specific loop where kr = minimum height required
R = Normal reaction force between track and particle
m = mass of object
a = acceleration towards the centre of the circle
θ = angle that the ramp makes with the horizontal
g = acceleration due to gravity
Ek = Kinetic energy
Ep = Gravitational potential energy
v = velocity at a given point