The Roller Coaster and the "Natural Path"

A classic example for HS physics teachers when discussing circular motion is the roller coaster loop. And the classic difficulty is getting students to understand the role the track plays in exerting force on the roller coaster car. To help with that, I have developed the idea of the "natural path".

After the basic discussions of circular motion and the emphasis that the force needed for circular motion  is not a real force, but the result of other real forces (I seldom call it centripetal force), we go to the roller coaster. However, I start with a typical loop cut in half vertically. We discuss what path a car would take when it came to the top of the loop and flew out into space. We get the typical projectile path. We draw one for a car moving at high speed and then another for one moving at low speed. I call those the "natural paths" that the car wants to follow.

I then draw in the rest of the coaster loop. We look at the high speed case and note that the natural path lies outside the loop. What prevents the car from following its natural path? The track. It must exert a force downward on the car to make it move in a circle.

We then look at the low speed case and see that the track must exert an upward force to keep it moving in a circle. We discuss what a rider would feel for each of these cases, with my in-class examples all starting with a roller coaster engineer specifying the force she wants a typical 75 kg rider to feel and then calculating how fast the car has to be moving to satisfy that requirement.

So, what will you feel?

Now comes the task of getting students to understand what the roller coaster passengers will feel. It starts with what they are feeling now. We talk about the forces they feel as they are sitting on the classroom stools. We decide that "normal" is when they feel the seat pushing up against them with a force equal to their weight. It usually takes some more discussion to get them to realize that they would feel "normal" when something is pushing against their butts, whether is it up or down. So, when the track (and thus the seat) is pushing with a force against your butt, you feel attached to that seat; you don't feel like you are falling. If that seat no longer pushes against your butt or is even pulling your butt down, you no longer feel attached to that seat; you feel like you are falling.

For advanced students

If you have students with a good calculus background, you can have them compare the curvature of the natural path to the curvature of the loop. And they can then calculate the force exerted by the track from the differential equation F=dp/dt.