We all know the feeling when a car takes a corner too fast and we are thrown outward by the centrifugal force. This effect is deliberately exploited, and accentuated, in designing rollercoasters: rapid twists and turns, surges and plunges thrill the willing riders.
Many modern rollercoasters have vertical loops that take the trains through 360 degree turns with the riders upside-down at the apex. These loops are never circular, for reasons we will explain.
Circular Vertical Loops
The dynamics of rollercoasters were reviewed in , where there are more details than we can present here.
Let’s first examine what happens for a vertical loop. If we ignore frictional losses, the mechanical energy is conserved, with equal values at the top and bottom of the loop:
The first term on each side is the kinetic energy, the second is potential energy. The notation is conventional or self-evident. Then
where is the radius of the loop. Thus, the centrifugal forces at top and bottom differ by , independent of loop size:
Suppose now that the speed at the top of the loop is , resulting in a centrifugal acceleration of . Since gravity opposes this, the nett acceleration is upward. At the bottom of the loop, the centrifugal acceleration is . Adding the acceleration due to gravity, the total -force downward is . This is more than most people could or would tolerate.
Clearly, circular loops are unsuitable for rollorcoasters. Many other loop shapes have been tried. It is easy to design loops for which the centrifugal force is constant. Loops with constant -force are another possibility, though they may not be so exciting for riders. We look at a third possibility, using clothoids.
Clothoids are segments of a spiral curve first studied by the great Swiss mathematician Leonhard Euler in 1744. Later, and independently, the spiral was re-discovered by the French physicist Marie Alfred Cornu when he studied the diffraction, or spreading of light passing through a narrow slit, and it is now known as the Cornu spiral.
A clothoid has curvature that increases linearly with distance along the curve. Suppose gives arc-length along the curve, and is the angle between the -axis and the tangent to the curve. Then the curvature is the rate of change of with , so we have
Now suppose the curvature varies linearly with arc-length, . Then, with obvious choice of initial values, and and are given by
These are standard integrals called Fresnel integrals. Plotting against we get the Cornu spiral, shown in the figure below.
We have seen that a circular loop might have a centrifugal force of or at its lowest point. If the circular arc is linked directly to a horizontal section, the onset of this force will be abrupt, and very uncomfortable. To avoid sudden changes, clothoids are used to ensure a gradual increase in centrifugal force. The use of clothoids for rollercoaster design was pioneered in 1975 by engineer Werner Stengel, founder of Stengel Engineering (Ingenieur Büro Stengel GmbH).The figure above shows a loop on a rollercoaster. (See also the figure at the top of this post). It is clear that the curvature is large at the top of the loops and smaller lower down. The centrifugal force is proportional to the curvature, so this ensures that the -force does not reach an unacceptable level.
 Pendrill, Ann-Marie, 2005: Rollercoaster loop shapes. Phys. Education, 40, 517-521. http://physics.gu.se/LISEBERG/eng/pe5601.pdf
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