The Physics Club: Roller-Coaster Designs & More...

Pi on my Cake

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In the Parks
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Rollercoaster Physics Part 3: Friction (Applied)

Alright, back to to science lol. So now that we know what the coefficient of friction is, we need to know how to find it and how to apply it. To do that, we need to understand how to draw a Free Body Diagram and some basic Newtonian physics (Don't worry, it is not as bad as it sounds).

First, we are going to start at equilibrium (it is neither moving not rotating). We are going to draw a Free Body Diagram. For simplicity, we will imagine the rollercoaster car full of people as simple square. Physicists are lazy, so we imagine basically everything as simple squares lol.
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Next, we start adding forces to our free body diagram. First is gravity. Fg is equal to mass times the accelration due to gravity (g=9.81 m/s^2).
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Next is the Normal force. When you are standing still, gravity doesn't just pull you down to the earth's core. The ground pushes back at you. N is the force of the ground pushing back. If you are on flat ground and there is no other forces in the y-direction (up and down), N is equal to Fg which is equal to m*g.
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So, how does this relate to friction? Well, when the object is in motion (in this case going to the right), another force comes into play. Friction Force (Ffr)! Ffr goes against the direction of motion. Ffr is equal to the Normal Force times the Coefficient of Kinetic Friction (Ffr=N*u)
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The total force on the object is equal to the sum of all the forces. On a flat surface, the Normal forces and the force of gravity cancel each other out (one pushes up, one pulls down). So, the only force we worry about is Ffr. Since the total force equals mass times acceleration (F=m*a) and mass doesn't change, We can relate acceleration to the coefficient of kinetic friction directly.
F=m*a=Ffr
Ffr=u*N=u*m*g
m*a=u*m*g
u=a/g
We know g (9.81). We can solve for a easily using the kinematic equations and simple experimentation of pushing stuff and seeing how long it takes to stop. The Coefficient of Friction doesn't change for the same two things rubbing against eachother, so once we know it for a simple, flat situation, we can apply it to more complex situations, like roller coasters!

Of course,everything needa to be designed with a factor of safety. There may be imperfections in the materials or broken parts welded back together (*cough* WDW Space Mountain *cough*) or all kinds of things that can change the coefficient of kinetic friction. Don't forget the effects of rain and stuff like that! Heavier guests will make the Normal Force larger which will make the friction force larger too. But, in general the Ffr found works well enough as long as we use a factor of safety.

So, now that we can calculate friction, we can estimate the energy lost (E_lost) from our energy equations earlier allowing us to do the calculations we need to figure out speeds and heights and all!
 

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