To me the scariest rides at the park are the roller coasters. The Big Dipper is similar to many of the roller coasters that have thrilled passengers for most of this century. The cars are pulled by chain to the top of the highest hill along the track. Released from the chain as the front car begins its descent, the unpowered cars have almost no speed and only a small acceleration. As more cars get onto the downward slope the acceleration increases. It peaks when all the cars are headed downward. The peak value is the product of the acceleration generated by gravity and the sine of the slope of the track. A steeper descent generates a greater acceleration, but packing the coaster with heavier passengers does not.

When the coaster reaches the bottom of the valley and starts up the next hill, there is an instant when their cars are symmetrically distributed in the valley. The acceleration is zero. As more cars ascend, the coaster begins to slow, reaching its lowest speed just as it is symmetrically positioned at the top of the hill.

A roller coaster functions by means of transfers of energy. When the chain hauls the cars to the top of the first hill, it does work on the cars, endowing them with gravitational potential energy, the energy of a body in a gravitational field with respect to the distance of the body from some reference level such as the ground. As the cars descend into the first valley much of the stored energy is transferred into kinetic energy, the energy of motion.

If the loss of energy to friction and air drag is small, the total of the potential and kinetic energies must remain constant throughout the descent and even throughout the rest of the ride. The coaster gains kinetic energy and speed at the expense of potential energy. If the first valley is at ground level, the transfer is complete, and for a moment the coaster has all its energy in the form of kinetic energy.

Without energy losses the coaster could climb any number of hills as high as the one from which it is released (but no higher). To be sure, friction and air drag do remove energy from the coaster, and its total energy content dwindles. It can no longer climb high hills, which is why the last stages of the track consist only of low hills.

The length of a ride on a roller coaster depends on the speed. If the ride is to be fast, the launching hill should be high so that the total energy is large. The rest of the track should be low so that most of the energy remains kinetic.

The choice of a seat on a roller coaster makes a difference in the ride. Some people prefer the front seat because the descent from the launching site presents the pleasingly frightening illusion of falling over the edge of a cliff. Other people prefer the psychological security of the rear seat.

The choice of a seat also determines the forces felt by the passenger. Consider the first descent. The front car starts down slowly because little of the coaster's energy is then kinetic. The speed of the cars increases as an exponential function of time, so that the rear car starts down at a much higher speed than the front car did. Although the passengers in the front car get an unobstructed view of the descent, the passengers in the rear car have a stronger sense of being hurled over the edge.

At the edge one force on the passenger is from the change in the direction of his momentum vector. Initially the vector is horizontal, but soon it points toward the valley. The force necessary to effect this change in direction is delivered by the safety bar or seat belt that keeps the passenger in the car. That force, which points downward and back toward the hill, is part of the thrill of the ride. A passenger in the rear feels the force more than a passenger in the front because the size of the force is proportional to the momentum, which is greater for the passenger in the rear.

The story is different in the valley. Again a force from the coaster is necessary to redirect the passenger's momentum. This time the momentum is initially downward toward the bottom of the valley and then is redirected toward the top of the next hill. The front passenger has a large momentum and is subjected to a large force. By the time the rear car reaches the bottom of the valley the movement of the front cars up the next hill has slowed the coaster. A passenger at the rear has less momentum and is subjected to a smaller force.

At the crest of the hill the passenger gets a force leveling his momentum vector. At the rear of the coaster the force can be considerable if the front is already well down the next slope. To a passenger at the rear who is loosely held in place by a safety bar a fast trip over a hill provides a brief sensation of being lifted from the seat. He arrives at the crest with a large momentum. Until he encounters the safety bar and is redirected he continues to travel upward even though the coaster has leveled out below him. The faster the coaster goes over a hill, the greater the sensation of being thrown free.

The brave passenger is one who rides the roller coaster without holding on. I tried this once while arriving at the crest of a hill at high speed. I avoided being thrown free of the coaster by catching my thighs on the safety bar at the last instant. Thereafter I kept a tight grip on the safety bar.

Roller coasters such as the Big Dipper have been around for more than 50 years. Recently a new type of coaster has appeared. The principles are seen in the Double Loop and the Corkscrew. The Double Loop at Geauga Lake begins like the Big Dipper in that a chain pulls the cars to the top of the first and highest hill. After the coaster travels over a few smaller hills (and before it loses too much of its energy to friction and air drag) it runs through two vertical loops. The ride is splendidly unnerving. In the times I managed to open my eyes while traveling through the loops I saw the world turn upside down, the ground race up toward me and the world turn upside down again.

The coaster on the Double Loop is held onto the track by a double set of wheels, one set on the top of the rails and the other set on the bottom. When the coaster is on the normal section of the track, its weight rests on the top set of wheels. When it is in the loop, the other set of wheels can come into play. They keep the cars from flying off the track.

When the coaster enters a loop, I sense three forces. One is my weight, which of course is directed downward. Another is the force from the seat. The third is the apparent centrifugal force downward, which seems to add to my weight; it makes me feel as though I am being pushed into the seat. At the top of a loop the apparent centrifugal force is upward, and I feel light.

The centrifugal force is a fiction. No outwardly directed force is at work. The notion of a centrifugal force is useful, however, since it easily explains what a passenger feels. The perspective of someone on the ground is more to the point: a combination of real forces causes the rider to move in a circle instead of a straight line.

If the circular motion is to be maintained, the net force must be toward the center of the circle. At the bottom of a loop the passenger's weight vector is downward (and therefore away from the center of the loop). An upward force acts on the passenger from the seat. Since the push from the seat is greater than his weight, the net force points toward the center of the loop and he begins the circular motion. From the passenger's perspective the large push from the seat is sensed as a centrifugal force pressing him into the seat.

At the top of the loop the forces have changed. The passenger's weight is still the same and is still pointed downward, that is, toward the center of the loop. The push from the seat is also downward. The two forces combine in the net force that makes the rider continue in the circle.

This time the force from the seat is smaller. One reason is that at the top of the loop the coaster has less kinetic energy and so is traveling slower. Moreover, the force from the seat is now augmented by the rider's weight vector instead of having to oppose it. The rider senses the force from the seat as a small centrifugal force.

How high must the coaster be at the start of its journey (with essentially no initial speed) if it is to have at the top of the loop the speed that will hold it firmly on the track? To answer the question I made two assumptions. The first was that the coaster has only one car. The second was that the energy losses from friction and air drag are negligible. With these assumptions I found that the first hill must be higher than the top of the loop by at least half the radius of the loop.

The first assumption is a convenient simplification. If the coaster is long, one must consider the rise and fall of its center of mass rather than considering only one car. Since only part of the coaster is at the top of the loop at any given instant, the center of mass never reaches that height, and so less energy is actually required to keep the coaster on the track than would be needed if there were only one car.

As for the second assumption, if the losses of energy were entirely negligible, the unpowered coaster would arrive at the loop with all the energy it got at the launch. The intervening hills and valleys would not matter. They do matter, of course, because they provide more opportunity for energy losses. Therefore the initial hill must be higher than the theory would indicate. On the Double Loop at Geauga Lake the initial hill is considerably higher than the theoretical value, so that the coaster is still traveling at a good clip when it reaches the top of the loop.

The Corkscrew is a similar roller coaster except that the loops are helical. Once the coaster enters the loops it moves in a corkscrew fashion until it emerges again. At two points the passengers are fully upside down.

The physics of this ride is similar to that for the Double Loop. The major difference lies in the direction of the apparent centrifugal force. With the Double Loop the center of the motion in a loop is at a single point. The centrifugal force appears to be directed radially outward from that point. As the coaster travels around the loop, this force rotates in a vertical plane. With the Corkscrew the center of motion continuously moves vertically and horizontally as the coaster travels through the loops. Hence the direction of the apparent centrifugal force is not confined to a vertical plane. This added feature is one reason the Corkscrew has become so popular with coaster addicts.

Geauga Lake has two other rides that are similar to the standard roller coaster. The water slide starts high above the ground. Water pours down the interior of the slide to provide lubrication and even a small amount of propulsion. The principle is simple: the initial gravitational potential energy is steadily converted into kinetic energy, so that the slider's speed increases during the descent. The lubrication provided by the water diminishes the loss of energy to friction.

The other ride is the Gold Rush Log Flumes. Passengers board a small boat shaped like a hollow log. It is really a car like the ones on the Big Dipper. Water flowing through the flume pushes the boat along until it is engaged by a chain system that drags it up a tall hill. From the crest the boat descends rapidly down a slope of about 45 degrees. At the foot of the hill it speeds into a trough of water, which quickly slows the motion and satisfyingly drenches the passengers. They also seem to be thrown forward, but the experience is illusory; what happens is that they continue to move forward briefly at the former speed.

Most of the other rides at an amusement park are based on rotational mechanics. The mildest of them is the merry-go-round. Here the rate of rotation is just enough to give the passenger a moderate sensation of centrifugal force. He seems to be thrust outward. Actually his body leans outward because the horse moves away from him as it travels in a circle and ends up pulling him along.

The Ferris wheel is similar except that its plane of rotation is vertical. The apparent centrifugal force seems periodically to increase and decrease the passenger's weight. When he passes through the bottom of the circle of travel, the centrifugal force appears to push him downward into the seat as if he then weighed more. In reality the seat pushes strongly against him as it keeps him moving in a circle. This force must be strong because it opposes the passenger's weight. At the top of the circle the passenger has the sensation of being somewhat lighter because the apparent centrifugal force is then upward, seemingly pulling him out of his seat. Actually the sensation comes from the fact that the force from the seat is then smaller.

At midpoint of the descent an even stranger sensation is felt. The force from the seat matches the passenger's weight, and the centrifugal force is outward. Hence the passenger feels as though he is about to be thrown forward out of the compartment.

My favorite among the rotating rides is the Rotor, which is a vertical cylinder with a diameter of about 12 feet. The rider stands with his back against the wall as the cylinder begins to spin. When the maximum spin rate is reached, the floor drops away, but the rider remains stuck to the wall. A particularly agile person might be able to squirm enough to get himself into an angled position or even upside down.

Why does the rider stick to the wall? From his perspective a centrifugal force pins him there. The resulting friction between him and the wall prevents him from falling when the floor is removed. A high rate of spin is called for, so that the apparent centrifugal force generates enough friction.

From the perspective of an outside observer the story is different. The rider is constrained to move in a circle because of a force from the wall. This centripetal force is responsible for the friction. Still, the spin rate has to be high if the force from the wall is to generate enough friction to hold the rider in place.

The Rotor at Geauga Lake has a roughly textured wall to increase the friction. With a smoother wall the centripetal force would have to be stronger to keep the rider from slipping. (One would either have to increase the spin rate or build a cylinder with a larger diameter.) Each time I rode the Rotor I was impressed by the overwhelming sensation that a centrifugal force was pushing me against the wall. In reality the wall was pushing on my back.

In order for the rider to stay in place his weight (a force vector downward) must not exceed the friction (a force vector upward). The amount of friction can at most be equal to the product of the friction coefficient (which depends on the roughness of the surfaces in contact) and the centripetal force from the wall. I estimated that the spin rate had to be about 30 revolutions per minute to hold me against the wall. Indeed, the apparatus did turn at about that rate.

Several other rides at Geauga Lake involve an apparent centrifugal force. The Muzek Express consists of a series of cars moving on a circular track that traverses several small hills. The diameter of the track is roughly 30 feet. The ride is fast, and so the centrifugal force on a passenger is quite strong. The hills provide extra thrills. Usually two people ride side by side in a car. Since they both feel an outward force, the passenger on the outside is squeezed against the wall of the car by the passenger on the inside. The forces are surprisingly large even if a passenger is small. I cannot avoid being pushed into the wall even when the inside passenger is my young daughter, who weighs less than half what I do.

The Enterprise is a rotating ride with cars individually suspended on radial arms extending from a central hub. As the cars begin to move in a horizontal circle the apparent centrifugal force makes the car rotate outward on the radial arm. Soon the car has rotated almost 90 degrees, and the passenger can see the ground directly below the window that originally was on the inside.

This rotation results from the way the mass of the car and the mass of the passenger are distributed with respect to the suspension axis of the car. A combined centrifugal force operates on the common center of mass of the passenger and the car. Initially this point lies below the suspension axis. Also acting through the point is the combined weight of the passenger and the car. These two forces compete in orienting the car. Initially gravity pulls the car into the normal orientation, but as the ride moves faster and the centrifugal force gets stronger the car is rotated increasingly out of the vertical.

This much of the ride was disturbing, but the next part almost did me in. Once the ride had reached its highest speed the large arm that held the central hub was turned to make the plane of the moving cars vertical. I was then moving in a vertical circle, being completely upside down at the top and greatly compressed by the forces acting on me at the bottom. I closed my eyes and began to count the prime numbers.

My next ride also held a surprise. It was a set of swings about 20 feet in diameter suspended from a central hub. When the hub began to turn I moved in a circle below the rim of the hub. As the speed increased, the apparent centrifugal force moved me outward so that I traveled in a larger circle than before The faster the hub turned, the larger the circle was.

From my perspective three forces affected me. I still had weight, which was directed downward. The chair and its suspension chains provided a second force directed toward the attachment of the chains to the overhead hub. The third force was the fictitious centrifugal force I felt throwing me outward. The angle between the chains and the: vertical was set by the balancing of the three forces. When the speed of the ride increased, the angle also increased, so that the forces again balanced.

The surprise of the ride was that the hub soon tilted about 10 degrees or so out of the horizontal. Part of my travel around the apparatus was then downhill. My speed increased during the descent as potential energy was converted into kinetic energy. As a result I circled the apparatus in a large radius. In the uphill part of the circle I slowed as the hub was forced to hoist me, and so here I circled with a smaller radius.

I ended my busy day at Geauga Lake with three rides that delivered similar types of motion. The first was the Scrambler, which has long been popular at amusement parks. It consists of a central hub from which several radial arms extend. I call them the primary arms. At the end of each primary arm four secondary arms extend outward. Each one carries at its outer end a car for two or three passengers.

The ride consists of two circular motions. The primary arms rotate steadily about the center of the ride while each set of- four secondary arms twirls below the pivot at the end of the primary arm. From an overhead perspective the primary arms move clockwise, the secondary arms counterclockwise. (In a related ride called Calypso both motions are clockwise.)

I set about studying the types of motion in rides such as the Scrambler and the Calypso. To model what happens to a rider I focused on a single primary arm (turning clockwise3 and a single secondary arm (turning in either direction). As the primary arm makes a full revolution does the passenger loop or spiral? Where are the speed and the acceleration greatest? How should the arms rotate to give an unforgettable ride? Should the arms be approximately the same length (as they are in the Scrambler and the Calypso)?

I found that if the arms are the same size and rotate at the same rate and in the same direction, the ride is bound to be rather boring, because the passenger merely goes in a large circle. The ride is not much better if the arms turn in opposite directions. In this arrangement the passenger would travel on a straight line over the center to the opposite side of the ride and then would return on the same line.

A better ride results when the primary and secondary arms rotate at different rates. Suppose the secondary arm turns twice as fast as the primary one. When the primary and secondary arms move in a clockwise direction, as they do in the Calypso, the passenger first spirals in toward the center of the ride and then out again, so that he travels through a loop on the side opposite to the starting point. After spiraling outward he passes through his initial location and begins the trip again.

His speed and acceleration are highest when he is farthest from the center, that is, when he passes through the initial point. They are lowest when he passes over the center of the ride. My calculations approximate the conditions of the Calypso but are off somewhat because to simplify matters I visualized arms of equal length. In order to accommodate all the primary and secondary arms on the Calypso the secondary arms are shorter than the ones in my calculations, so that the cars do not crash near the center of the ride.

If the primary and secondary arms turn in opposite directions, as they do in the Scrambler, a more interesting motion results. At first the passenger moves counterclockwise, but he quickly heads for the center of the ride and then outward again. When the arms are fully extended, he is directed back toward the center. When the primary arm has completed one revolution, the passenger has traveled through a pattern resembling three narrow petals. He is moving at the highest speed when he passes over the center of the ride. Surprisingly, the acceleration there is the lowest. The speed is the lowest and the acceleration the highest when the passenger is farthest from the center.

The low speed at that point results because the circular motion of the secondary arm is carrying the passenger counterclockwise while the motion of the primary arm is clockwise. The two motions oppose each other when the passenger is farthest from the center and augment each other as he is carried close to the center. The high acceleration at the far point develops because the direction of the velocity out there is changing rapidly.

On my home computer I worked out the paths for other conditions. If the primary arm is much longer than the secondary arm, the passenger might spiral toward and then away from the center. In other cases he would move through a path consisting of a series of cusps or loops superposed on a large circle. Another interesting situation arises when the secondary arm is longer than the primary one. If the secondary arm turns slower than the primary one and in the same direction, the passenger will spiral gradually toward the center and then away from it. If the motions are in opposite directions, the ride will include an abrupt change in direction.

My last ride of the day was on the Tilt-A-Whirl. I sat in a compartment that was free to move around a small circular track, pivoting on a center point at my feet. Six of these compartments, running on a hilly track, moved around the center of the apparatus while they were also pivoting on their individual center points. The result was three types of motion. One was a basic counterclockwise circling about the center of the entire ride. The second was a smaller circling of the compartment in either direction. The third was the vertical motion over the hills.

The interesting part was that I could often control the small circling of my compartment by anticipating the hills and shifting my weight. When the compartment was turning in its small circle and beginning to descend from a hill in the larger motion, I threw my weight in the direction of the compartment's turn. I was transforming some of the potential energy of my body (from being up on the hill) into kinetic energy applied to the rotation of the compartment.

When I timed this exercise correctly, I set the compartment spinning rapidly. My experience was similar to that in the preceding two rides. If the spin direction was in the same direction as the large-scale circling of the ride, the speed and centrifugal force were quite large when I was farthest from the center of the ride. If the spin was in the opposite direction, the acceleration was high when I was far from the center but the speed was high closer to the center.

You might explore the amusement parks near you for other rides. One that I have heard described, but lack the courage to even look at, is Demon Drop. The victim—sorry, the passenger—is secured in a chair that is lifted 131 feet and then dropped in a virtually free fall to the ground. The huge kinetic energy of the ride is apparently dissipated when the apparatus curves into a horizontal section of track at the bottom of the fall. I have no intention whatever of verifying this assumption.

Bibliography

HARRY G. TRAVER: LEGENDS OF TERROR. Richard Munch. Amusement Park Books, Inc., Mentor, Ohio, 1982.

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