Question

Not a question , but as a kid i always wondered how these seriously work. There must be a lot of calculations involved to construct such rollercoasters. And , my question is , how do they calculate something when there is air-resistance and all that stuff involved. And is radius of curvature applied here , because till now i have never found any application of these sub-topic

Answer 1

http://www.youtube.com/watch?v=9yFvlcw8YtY

Answer 2

Hi! I only watched little bits of the YouTube video, but that's interesting!

I'm not highly mathematical, unfortunately. That's a work in progress.

I bet that a lot of basic kinematics can be applied, though!

For air resistance, I'm sure that they identify sections where the roller coaster is slow and air resistance is negligible. The force due to air resistance can be expressed as the velocity multiplied by a drag coefficient (a constant). The coefficient then just gets convenient units. For some shapes and I think higher speeds, velocity is more considerable, and gets an exponent to make it's effect on the drag force more significant. Units of the drag coefficient change conveniently. :P

You mentioned curvature, too! I'm sure this is taken into account. Curvature is the reciprocal of the radius. And centripetal force is dependent upon this! You could say that the curvature is an independent variable of a centripetal force function. The centripetal force is part of the net force that prevents the train of cars from a straight-line trajectory. This is very considerable on loops, helixes ... any turn! Even the spots where the roller coaster is headed down and then has to become level. All of this centripetal force must be enacted on the train by the tracks! So, those tracks have to be reinforced specially to deliver this centripetal force.
For example, a helix might have supports that don't extend just below the track, but also to the side - outside of the turn - for reinforcement.

I would guess that supports be more frequent along the track in these areas.

Answer 3

Going back to air resistance, trains might be shaped to minimize air resistance (heavily relying on the shape of the front car, I'd guess).

Equations might look like:
\(\vec F_{\sf drag}=-\vec vc\qquad\quad c>0\)
and
\(F_{\sf centripetal}=m\dfrac{v^2}r=mv^2\kappa\)

Answer 4

To be honest, I'm not sure if that curvature (\(\kappa\)) thing is accurate. I'm working on a rusty memory.

Answer 5

But , air resistance maynot be constant

Answer 6

Right! People like to, when possible, simplify air resistance to depend on the velocity and some constant.

So a fast moving object will experience higher resistance. You can feel this with a qualitative experiment with two steps:
1. put your hand out and walk
2. put your hand out and ride really fast on a bike, train, car, etc

You'll feel the difference because your hand is pushing more air every second! Giving the air that kinetic energy, work must be done by your force. It's greater at higher speed.

The coefficient of drag is like a catchall. So, it tries to account for the shape of an object, or the shapes it will be as it falls. Also, it tries to capture the texture of the object. And air pressure swells. And, REALLY, everything that isn't the velocity.
The coefficient of drag is a catchall when it's experimentally determined and we don't have to put any thought into it :)

Answer 7

interesting