The loop that is not a circle

Turning is an acceleration, and constant speed does not help found that anything moving on a curved path is accelerating towards the inside of the curve, at its speed squared times the curvature, and that something must supply the force. On a roller coaster that something is the track, pushing on the car’s wheels, and the car pushing on the riders through their seats. What the riders feel is that push, and its size and its suddenness decide whether a ride is thrilling or injurious. A vertical loop is the most demanding piece of track there is, because the car must be travelling fast enough at the top to stay on — and everything it gained to get there it carries back at the bottom.

The first looping coasters, in the 1890s, used circles, and their riders came off with sore necks; within a few years the loops had gone, and coasters stayed loop-free for most of the twentieth century. When loops came back, in 1976, they were a different shape. The physics of why can be worked out with nothing but the conservation of energy and the geometry of curvature, and it turns out to have been solved for a different purpose — keeping trains on their rails — eighty years before.

Six g that no circle can avoid #

Take a circular loop first, and ask how fast the car must enter it. At the top, upside down, the car needs enough speed that the curve’s demand for centripetal acceleration, $v^2/R$, is at least $g$, so that gravity alone does not pull it off the track faster than the curve bends away; any extra the track supplies by pushing down. With $v^2 = gR$ at the top the seat force there is zero, the riders weightless. Coming down the loop to the bottom, $2R$ lower, the car gains speed: by conservation of energy, as the hill that gives it back found for any frictionless track, $v^2$ grows by $2g \times 2R$, to $5gR$. At the bottom the track must supply the centripetal force $mv^2/R = 5mg$ and hold up the car’s weight $mg$ besides: a seat force of six times the rider’s weight.

The striking thing is that $R$ has cancelled. A small loop and a large loop entered at the speed that just keeps the car on the track give exactly the same six g at the bottom. The third figure shows the general statement for circles of four sizes: whatever the speed at the top, the force at the bottom is exactly six g more. The climb from bottom to top takes $4gR$ out of $v^2$, the curvature $1/R$ turns that into four g less centripetal acceleration, and gravity, adding to the seat force at the bottom and subtracting from it at the top, accounts for the other two. A circle’s size changes how fast its riders travel, not what they feel.

Six g is a great deal. Fighter pilots train to tolerate more, briefly, with special suits and breathing techniques; an untrained passenger subjected to six g in a seat for a second or more can lose vision and, with the head unsupported, strain the neck. Modern roller coasters are designed to stay well below that almost everywhere on their track. A circular loop, entered fast enough to be safe at the top, cannot.

Why the shape of the track cannot change the speed #

The six-g argument used only two facts, and both deserve a second look because the clothoid depends on them too. The first is that the car’s speed at any height is fixed by how far it has fallen, whatever the shape of the track between. The track pushes on the car only at right angles to its motion, so it does no work, and the floor that does no work found what follows: the force of the track can change the car’s direction as much as it likes, but only gravity changes its speed. A circle, a clothoid and any other curve of the same height deliver the car to the top at the same speed for the same entry speed. The shape decides only how much the track must push to turn the car, not how fast the car goes.

The second fact is that the riders feel only the track’s push. In the frame of the car, the riders seem to be pressed outward from the centre of the curve by a force they did not apply, which the forces that are not there identified as the centrifugal force of a rotating frame — a book-keeping term, not a force any object exerts. From the ground the account is simpler: the seat pushes the rider round the curve, and the rider feels the seat. At the top of a loop entered at the minimum speed, gravity alone provides exactly the centripetal acceleration the curve requires, the seat pushes not at all, and the rider floats.

The same arithmetic has an edge. If the car is slower than the minimum, the curve requires less than gravity supplies, the seat would have to pull rather than push, and an unrestrained car would leave the track — exactly as the ladder that lets go of the wall found a sliding ladder leaving its wall at the moment the wall’s push would have to become a pull. Modern coaster cars carry wheels below the rail as well as above it, so they cannot leave; the minimum speed then becomes a comfort limit rather than a safety one.

The independence of the six g from the loop’s size has a family resemblance worth noticing. The jump that does not get higher with size found that animals of very different sizes jump to roughly the same height, because both the energy their muscles can deliver and the energy needed to rise scale with their mass. In the loop, both the speed squared the car gains and the curvature that converts it into force scale with the loop’s radius, in opposite directions, and the force comes out the same at every size. In each case a dimensionless ratio is fixed by the geometry, and size drops out.

The jolt at the entry #

The second problem with the circle is worse than the first. A straight track has zero curvature; a circle has curvature $1/R$; join them, and at the joint the curvature jumps. The centripetal acceleration is speed squared times curvature, so it jumps too, and the riders, who were feeling one g on the straight, are suddenly feeling six and a half. The fourth figure draws it against time as a step. The rate at which acceleration changes, the jerk, is what a body feels as a jolt, and at a step it is limited only by how gradually the car’s wheels, spread over a few metres, take up the new curvature. Riders of the Flip Flap Railway, built at Coney Island in 1895 with a circular loop about eight metres across, complained of exactly that whiplash in their necks, and the ride closed within a few years.

A curve whose curvature grows with distance #

The cure is to let the curvature start at zero and grow. The simplest curve that does that is one whose curvature increases in proportion to the distance travelled along it, $\kappa = s/A^2$ for some constant $A$. It is called a clothoid, or an Euler spiral, after Leonhard Euler, who studied it in 1744 as the shape of a spring bent by a force at its end. A car entering such a curve from a straight at constant speed feels its centripetal acceleration rise steadily from zero, with a finite jerk instead of a step.

A loop built from clothoids — the shape Werner Stengel designed for the Revolution at Six Flags Magic Mountain in California in 1976 — uses a clothoid on each side, from the straight up to the vertical, joined over the top by a tighter arc. The first figure draws one beside a circle of the same height. It is narrower, and tighter at the top: its top radius is 8.4 metres where the circle’s is 10. That tightness at the top is the second half of the trick. A smaller radius at the top needs less speed to keep the car on, so the car can enter the loop slower — 22.7 metres per second against 23.2 here for the same half a g at the top — and arrives at the bottom of the curve, where the curvature is gentle, with less speed to turn.

What the rider feels instead #

The second figure follows the seat force round both loops, entered at speeds that leave half a g at the top. On the circle the force is at its worst at the very start, six and a half g, and falls from there. On the clothoid loop it starts at one g, rises smoothly as the curvature grows, peaks at about four and a third g a quarter of the way round, and falls; the peak comes where the product of the speed squared, still high, and the curvature, now significant, is largest. Over the top, where both loops are circular arcs, the forces are the same — on any circular arc the force depends only on the speed at the top and the angle, not on the radius, which is the same arithmetic that made the six g universal.

The clothoid loop is not the end of the design problem. Modern coasters shape the whole track by specifying the forces they want the rider to feel at each moment, and then integrating to find the track that produces them, with the riders’ heads and the cars’ wheels as constraints. But the Euler spiral is the first step of that idea, and it is still the curve used at the start and end of every loop.

The same spiral in every bend #

The clothoid did not come from roller coasters. Railways discovered the problem first. A train running at speed from a straight track onto a circular curve receives the full sideways acceleration of the curve at once, and its wheels strike the outer rail at the joint; the blows damaged track and rolling stock and made the ride rough. Through the nineteenth century railway engineers introduced transition curves to ease the change, first using cubic parabolas, which approximate the clothoid near its start, and later the clothoid itself. Highway engineers adopted the same transitions in the twentieth century, because a driver following a road whose curvature jumps must turn the steering wheel instantly from straight to its full angle, which no driver does, and the car drifts towards the outside of the bend.

The last figure shows the effect on a road bend. Straight into a 200-metre arc at ninety kilometres an hour, the sideways acceleration appears all at once. Through a sixty-metre clothoid it builds up over two and a half seconds, and the steering wheel turns at a steady rate. The loop of a roller coaster is this railway curve stood on end, and the same constant that sets how fast the curvature grows sets how hard the jolt is.

The hill that pays for the loop #

Every metre per second of entry speed is bought with height, because a coaster car has no engine once it leaves the lift hill. Entering a twenty-metre loop at 23.2 metres per second needs a drop of $v^2/2g$, about 27 metres, before any losses; the clothoid loop’s 22.7 metres per second needs about 26. The difference looks small, but it is paid for twice. The lift hill must be taller by that much, and every subsequent element of the ride inherits the extra speed, with forces that grow as its square. The push that pays only for the hill found that lifting a load costs only the height gained, whatever the path; a coaster’s lift hill is that payment made once, and the rest of the ride spends it.

That is why the shape of the loop’s top matters so much to a designer. A circle must be entered fast enough to clear its large top radius, and the riders pay for that speed at the bottom. A clothoid loop, tight at the top, can be entered slower; the riders feel less at the bottom, the ride needs a lower lift hill, and the energy saved can be spent elsewhere — on a second loop, or a longer ride. The physical saving is in the curvature at the top, and the comfort saving is in the curvature at the bottom; one curve buys both.

The spiral that light also follows #

The Euler spiral turns up somewhere apparently unrelated, and the connection is worth seeing. The spiral that says how much light arrives found the Cornu spiral, the curve whose two coordinates are the Fresnel integrals, describing how much light reaches a point near the edge of a shadow: the light from each strip of the wavefront adds a small arrow, the arrows turn at a rate growing with distance from the point, and the tip of the chain of arrows traces a spiral. That spiral is the clothoid. Its defining property — direction changing at a rate proportional to distance along it — is the same whether the “distance” is along a railway track or across a wavefront, and the same integrals that tell an optician how a shadow’s edge is fringed tell an engineer where to lay a rail.

That is not a coincidence of names. Both problems ask for a curve built by adding up small straight pieces, each turned by an angle that grows linearly. Any phenomenon described that way — the bending of an elastic strip by a force at its end, as Euler had it, or the phase of a wave as the path length grows quadratically — produces the same spiral.

Where the model is simple #

The figures treat the car as a point sliding without friction along a curve in a vertical plane, entering the loop at a fixed speed. Real coaster cars lose energy to rolling resistance and air drag, of order ten per cent over a loop, which must be made up by a higher entry speed. The car is not a point; its wheelbase spreads it over a few metres, and riders at its front and back feel different forces as it enters and leaves a curve. The track twists sideways as well as bending vertically, because entry and exit must be offset so that the track does not run into itself, which the two-dimensional drawings show as a crossing. And a rider is not rigidly attached to the seat; the head, on a flexible neck, responds to jerk with a lag, which is what makes jerk rather than acceleration the quantity that injures.

The rider the forces do not include #

The figures show forces and shapes; they do not show the human response, which depends on the direction of the force relative to the body, its duration and its onset, and is known from experiments with volunteers in centrifuges and from accident records rather than from any calculation. Six g along the spine for a fraction of a second is tolerable for most people; the same force sustained, or applied sideways to an unsupported head, is not. And they show a single clothoid design; the real optimisation of a loop trades off peak force, entry speed, height and the length of track against one another, and different designers have settled on different shapes.

Still open: how much jerk the body can take #

Engineering standards for amusement rides specify limits on acceleration in each direction and for each duration, built up from decades of experience, but jerk — the rate at which acceleration changes — is much less well understood as a cause of discomfort and injury. Measurements of head and neck motion in riders, and models of the cervical spine’s response to sudden changes of force, suggest that jerk matters most for the neck and acceleration for the circulation, but how the two combine, and how they should be weighed against each other in a design, is still debated, and modern rides are tested with instrumented dummies as well as calculated.

The habit worth carrying away is to ask how a force arrives as well as how large it is. A circular loop entered fast enough to hold its riders at the top presses them at six g more at the bottom, whatever its radius, and steps the force on at the entry in an instant; a clothoid, whose curvature grows in proportion to distance, starts the force at one g and ramps it up, and with a tighter top lets the car enter slower, so a 20 m loop peaks at 4.3 g instead of 6.5. The same spiral eases every railway and road into its bends, and draws the edge of every shadow.