Roller Coasters and Amusement Park Physics PDF

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Lecture Topic: Roller Coasters and Amusement Park Physics People are wild about amusement parks. Each day, we flock by the millions to the nearest park, paying a sizable hunk of money to wait in long lines for a short 60-second ride on our favorite roller coaster. The thought prompts one to consider what is it about a roller coaster ride that provides such widespread excitement among so many of us and such dreadful fear in the rest? Is our excitement about coasters due to their high speeds? Absolutely not! In fact, it would be foolish to spend so much time and money to ride a selection of roller coasters if it were for reasons of speed. It is more than likely that most of us sustain higher speeds on our ride along the interstate highway on the way to the amusement park than we do once we enter the park. The thrill of roller coasters is not due to their speed, but rather due to their accelerations and to the feelings of weightlessness and weightiness that they produce. Roller coasters thrill us because of their ability to accelerate us downward one moment and upwards the next; leftwards one moment and rightwards the next. Roller coasters are about acceleration; that's what makes them thrilling. And in this part of Lesson 2, we will focus on the centripetal acceleration experienced by riders within the circular-shaped sections of a roller coaster track. These sections include the clothoid loops (that we will approximate as a circle), the sharp 180-degree banked turns, and the small dips and hills found along otherwise straight sections of the track.

The Physics of Roller Coaster Loops The most obvious section on a roller coaster where centripetal acceleration occurs is within the so-called clothoid loops. Roller coaster loops assume a tear-dropped shape that is geometrically referred to as a clothoid. A clothoid is a section of a spiral in which the radius is constantly changing. Unlike a circular loop in which the radius is a constant value, the radius at the bottom of a clothoid loop is much larger than the radius at the top of the clothoid loop. A mere inspection of a clothoid reveals that the amount of curvature at the bottom of the loop is less than the amount of curvature at the top of the loop. To simplify our analysis of the physics of clothoid loops, we will approximate a clothoid loop as being a series of overlapping or adjoining circular sections. The radius of these circular sections is decreasing as one approaches the top of the loop. Furthermore, we will limit our analysis to two points on the clothoid loop - the top of the loop and the bottom of the loop. For this reason, our analysis will focus on the two circles that can be matched to the curvature of these two sections of the clothoid. The diagram at the right shows a clothoid loop with two circles of different radius inscribed into the top and

the bottom of the loop. Note that the radius at the bottom of the loop is significantly larger than the radius at the top of the loop. As a roller coaster rider travels through a clothoid loop, she experiences an acceleration due to both a change in speed and a change in direction. A rightward moving rider gradually becomes an upward moving rider, then a leftward moving rider, then a downward moving rider, before finally becoming a rightward-moving rider once again. There is a continuous change in the direction of the rider as she moves through the clothoid loop. And as learned in Lesson 1, a change in direction is one characteristic of an accelerating object. In addition to changing directions, the rider also changes speed. As the rider begins to ascend (climb upward) the loop, she begins to slow down. As energy principles would suggest, an increase in height (and in turn an increase in potential energy) results in a decrease in kinetic energy and speed. And conversely, a decrease in height (and in turn a decrease in potential energy) results in an increase in kinetic energy and speed. So the rider experiences the greatest speeds at the bottom of the loop - both upon entering and leaving the loop - and the lowest speeds at the top of the loop. This change in speed as the rider moves through the loop is the second aspect of the acceleration that a rider experiences. For a rider moving through a circular loop with a constant speed, the acceleration can be described as being centripetal or towards the center of the circle. In the case of a rider moving through a noncircular loop at non-constant speed, the acceleration of the rider has two components. There is a component that is directed towards the center of the circle (ac) and attributes itself to the direction change; and there is a component that is directed tangent (at) to the track (either in the opposite or in the same direction as the car's direction of motion) and attributes itself to the car's change in speed. This tangential component would be directed opposite the direction of the car's motion as its speed decreases (on the ascent towards the top) and in the same direction as the car's motion as its speed increases (on the descent from the top). At the very top and the very bottom of the loop, the acceleration is primarily directed towards the center of the circle. At the top, this would be in the downward direction and at the bottom of the loop it would be in the upward direction.

Force Analysis of a Coaster Loop The inwards acceleration of an object is caused by an inwards net force. Circular motion (or merely motion along a curved path) requires an inwards component of net force. If all the forces that act

upon the object were added together as vectors, then the net force would be directed inwards. Neglecting friction and air resistance, a roller coaster car will experience two forces: the force of gravity (Fgrav) and the normal force (Fnorm). The normal force is directed in a direction perpendicular to the track and the gravitational force is always directed downwards. We will concern ourselves with the relative magnitude and direction of these two forces for the top and the bottom of the loop. At the bottom of the loop, the track pushes upwards upon the car with a normal force. However, at the top of the loop the normal force is directed downwards; since the track (the supplier of the normal force) is above the car, it pushes downwards upon the car. The free-body diagrams for these two positions are shown in the diagrams at the right. The magnitude of the force of gravity acting upon the passenger (or car) can easily be found using the equation Fgrav = m*g where g = acceleration of gravity (9.8 m/s2). The magnitude of the normal force depends on two factors - the speed of the car, the radius of the loop and the mass of the rider. As depicted in the free body diagram, the magnitude of Fnorm is always greater at the bottom of the loop than it is at the top. The normal force must always be of the appropriate size to combine with the Fgrav in such a way to produce the required inward or centripetal net force. At the bottom of the loop, the Fgrav points outwards away from the center of the loop. The normal force must be sufficiently large to overcome this Fgrav and supply some excess force to result in a net inward force. In a sense, Fgrav and Fnorm are in a tug-of-war; and Fnorm must win by an amount equal to the net force. At the top of the loop, both Fgrav and Fnorm are directed inwards. The Fgrav is found in the usual way (using the equation Fgrav = m*g). Once more the Fnorm must provide sufficient force to produce the required inward or centripetal net force....