Importance Of Roller Coasters

“Why is the shape of roller coaster loops not circular?”

Introduction: Most of us have been on a roller coaster or at least seen one from a distance. I personally find the vertical loop section the most intriguing section. Surprisingly, none of the vertical loops are actually circular in shape. Instead, it is an almost oval shape. Not to mention, this design was the demand of two very important aspects, namely; safety and thrill. The explanation requires the use of theories of circular motion. Let us take an example of a stone tied to the end of a string. We know that when the stone rotates vertically, the tension in the string is maximum at the bottom of the loop. Which implies that the stone is apparently heavier at the bottom. Similarly, passengers

In case of a circular loop the coaster would have to begin the climb with enough kinetic energy so that it can, not only make it to the top, but have enough energy remaining in order to complete the loop without coming to a halt at the top.

The diagram represents an almost identical situation to that of a roller coaster. Considering the diagram and assuming that the roller coaster starts at the beginning of its ride with zero velocity, which it does, we have that the velocity at any instant equals : m×g×∆h=1/2 m×v^2 - 1

Since the mass is not changing, simplifying gives: g×∆h=1/2 v^2 -2

Rearranging in terms of velocity gives: v^2=2 g×∆h -3

From the equation of circular motion F = (mv^2)/r we can get that a=v^2/r -4

Hence, using this definition of acceleration we have: a=(2 g∆h)/r -5

This implies that the height at which centripetal acceleration is equal to the gravitational acceleration is ∆h=r/2