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(23b) Loop-the-Loop


22c. Flight (1)

22d. Flight (2)

23. Inertial Forces

23a. The Centrifugal Force

  23b. Loop-the-Loop

  24a.The Rotating Earth

24b. Rotating Frames

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(Optional Addition)

In the preceding section the motion of the "loop the loop" roller coaster was handled using the centrifugal force. You can also view this problem from the point of view of the outside world, using the centripetal force, but it is not as easy. At point A, on the top of the loop, both gravity and the centripetal force point downwards. So what is there that can keep riders in their seats?

Let us try solve that motion, using the concept of the centripetal force. A car going around a loop, with radius R and velocity V, is accelerating at a rate of V2/R towards the center (as long as it stays on the rails), and is therefore subject to a centripetal force mV2/R, also directed to the center. When the car is at point A, that force points downwards. Let "down" be now be taken as the positive direction along the vertical axis.

The centripetal force is provided by two sources: the weight mg of the car, directed downwards, and the reaction FR of the rails. We have at point A

    mg + FR = + mV2/R
    FR = + mV2/R – mg
where a positive FR pushes the car down, a negative one pulls it up

Now the car rides on rails. At point A the rails are above the car and therefore it can only push up against them. The rails then, reacting to the force, must push it down, somewhat similar to the situation in "Objects at Rest", in section #18 on Newton's second law. Thus FR must be positive: if it were negative it would mean that the rails were pulling the car upwards, which they cannot do.

We thus require FR > 0, that is

    mV 2/R – mg > 0
or, after adding mg to both sides
    mV 2/R > mg
This is the same result as was obtained using the centrifugal force: the problem can be solved in the outside frame of reference--but the process is a bit more complicated. The intuitive meaning is shown in the drawing.
  • If all forces on the car ceased at point A, it would
       continue along a straight line to point B,
        in accordance with Newton's first law.
  • If only gravity acted, it would follow a parabola
        to point C.
  • For the rails to exert a positive pressure,
        they must constrain the car to a tighter curvature
        than gravity alone, forcing it to move to point D.

Exploring further

    Actual looping roller coasters do not follow a circular path, but a "clothoid curve" (look up the term!!) which curves more tightly at the top and more gently at the bottom (you can see this in the illustration on the previous page). The tight curvature on top helps keep the passengers pushed outwards, and the gentle one at the bottom reduces the downward force on them, in sections where the centrifugal force and gravity add up in approximately the same direction.

        A short article on the Clothoid curve, linked to the Italian version of this web collection, was translated to English by Dr. Giuliano Pinto and after some further editing was incorporated into this collections: click here.

Next Stop: The Rotating Earth
Next Stop: #24b Rotating Frames of Reference

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Last updated: 9-22-2004
Reformatted 25 March 2006