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Transcript
Thrill U. THE PHYSICS AND MATHEMATICS OF AMUSEMENT PARK RIDES Physics © Copyrighted by Dr. Joseph S. Elias. This material is based upon work supported by the National Science Foundation under Grant No. 9986753. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Dorney Park/Kutztown University Thrill U. Introduction The Lehigh Valley is rich in tradition, culture and beauty. We are most fortunate to have a community of fine people who are dedicated to the enhancement of our quality of life. To these ends, Dorney Park and Kutztown University have collaborated in the development of an educational experience that will benefit the children of the Lehigh Valley and beyond. We call it Thrill U. Our goal is to provide a stimulating and challenging exploratory experience for high school students. We utilize some of Dorney Park’s best attractions in ways that promote a deeper and more profound understanding of select scientific and mathematical principles. Students are given the opportunity to examine and study relationships between the dynamics of the mechanical universe and the unique, structural features of the rides. Kutztown University of Pennsylvania is pleased once again to participate in a collaborative project that engages future teachers in serious work with leading educators and the community. For science and mathematics teachers, this represents the best of two worlds, a living classroom replete with experiential activities and a forum for examining the connections between theory and practice. For Dorney Park, this is yet another opportunity to showcase their outstanding amusement park. All who participate will examine the extraordinary structural design process that went into the construction of these fabulous rides. We extend to you the opportunity to examine our laboratory manual, review procedural aspects, and participate in our annual Thrill U. day that will be held on May 12, 2017. Thousands students from regional schools have participated in our annual Thrill U. With the addition of new and exciting activities, we believe that you and your students will find the day both thrilling and enlightening. Dr. Joseph S. Elias Professor Emeritus, Science Education College of Education Kutztown University of Pennsylvania Thrill U. Table of Contents Planning Team Page i Tips for Teachers Page ii Things to Bring Page iii The Rides Pages Apollo 2000 The Antique Carrousel The Ferris Wheel The Enterprise Revolution The Dominator The Sea Dragon White Water Landing The Scrambler The Wave Swinger Steel Force Energy Curves for Steel Force Centripetal Force and Steel Force The Talon Thunderhawk The Hydra Interpreting Graphs Possessed Demon Drop Meteor 1-123 Page 1 Page 8 Page 13 Page 19 Page 23 Page 27 Page 32 Page 37 Page 42 Page 47 Page 52 Page 61 Page 65 Page 67 Page 75 Page 79 Page 95 Page 104 Page 116 Page 124 Thrill U. Dorney Park/Kutztown University Planning Team The making of an event of such monumental scope can only be accomplished when the “players” are truly dedicated to its goals. Such is the nature of our planning team. The planning process began in August of 1997. Since then teams of science and mathematics teachers and students have enthusiastically participated in all phases of development. The professional staff of Dorney Park has graciously opened their doors, extending their guidance and technical support to those who developed the laboratories. One park professional likened it to a magician “revealing” well-kept secrets. The faculty, students, and administrators of Kutztown University have made the commitment of their time, energy and enthusiasm. Our goal has been and always will be academic excellence. We recognize the value of Thrill U. as an instrument befitting this goal. The combined efforts of all represent the true spirit of education and service. Acknowledgment Mr. William Landis Mr. Patrick Callahan Mr. Bernie Bonuccelli Mr. Joseph Greene Mr. Keith Koepke Mr. Edward Anthony Mr. Brent Ohl Ms. Carole Wilson Dr. David Drummer Mr. Richard Button Dr. Kathleen Dolgos Dr. Joseph Elias Dr. Deborah Frantz Dr. Neal Shea Ms. Brenda Snyder Mr. Glenn Frey Mr. Jeffrey Wetherhold Mr. Jeffrey Bartman Ms. Brandi Murphy Mr. Gerry Farnsworth Mr. Robert Guigley Ms. Maggie Woodward Allentown School District Delaware Regional School District Dorney Park of Allentown Dorney Park of Allentown Dorney Park of Allentown East Penn School District East Penn School District East Penn School District Kutztown Area School District Kutztown University of Pennsylvania Kutztown University of Pennsylvania Kutztown University of Pennsylvania Kutztown University of Pennsylvania Kutztown University of Pennsylvania Kutztown University of Pennsylvania Northwestern Lehigh School District Parkland School District Parkland School District Parkland School District Parkland School District Reading Area School District Upper Perkiomen School District i Thrill U. …and to the many graduate students of the Kutztown University of Pennsylvania who contributed to the development of this manual. Tips for Teachers To help make your day at the park more enjoyable, we have created a list of “tips for teachers.” Hopefully, this list will guide you through the pre-visit planning stage and answer some of your questions. Please don't forget your equipment, supplies and laboratory manuals. You may find that a camcorder might be functional in a variety of ways. Perhaps you wish to discuss the dynamics of the rides as a review, incorporate them within a laboratory practical, use as introductory preparation for next year’s trip to the park. You and your students should decide on which of the many rides you want to explore. Carefully peruse the complete list of activities and find those rides that will best benefit your students. Some rides may take more time than others to complete. You may find it necessary to ride several times on some of the rides in order to collect good data. As much as is feasible, introduce to the students the concepts to be studied and rides that you have chosen during the weeks leading up to the event. Plan time in class for calculations and analysis during the days following the experience. Each teacher needs to decide how the students from his/her school will complete the data collection sheets, and any other information, that her/ his students may need. We recommend that teachers in charge advise students who may be fearful of some rides, that riding is optional and not mandatory. Kutztown University students will serve as general assistants to the teachers. They will be stationed at each listed ride and the reserved pavilion from 10:00 AM until 3:00 PM. Follow the park map to the pavilion site and look for the Thrill U. banner. Inform your students that they may ask the university students any questions related to the event with the exception of specific questions that may be contrary to your objectives. Further information may be obtained by contacting: Mr. Matt Stolzfus Dorney Park 610.391.7607 [email protected] ii Thrill U. Dr. Joseph S. Elias Kutztown University of Pennsylvania [email protected] Things to Bring To make your day at the park as functional and enjoyable as possible we suggest that you arrange to bring some or all of the items listed below: tickets for you, your students and your chaperones copies of the activities that you and your students plan on doing stopwatches calculators clipboards paper and pencils masking tape protractors accelerometers inclinometers CBL (calculator based laboratory) if you have them and low range sensors for acceleration appropriate clothing with a change of clothing sunscreen, hats, raincoats money for food, drinks or phone measuring tape or string backpacks or plastic bags to keep laboratory manuals and equipment dry and together a good reserve of energy and enthusiasm for exploration Dorney Park Information For general information call (800) 551-5656 (610) 395-3724 Group Sales Information (610) 395-2000 or Matt Stoltzfus at 610.391.7607 with any specific questions about ticket sales for Thrill U.. iii Thrill U. or visit our website: www.dorneypark.com Thank you. ii Apollo 2000 Introduction: Rotational motion is a topic in physics that looks at objects that rotate or revolve around a point. This point is called the point of rotation. These objects have many properties associated with them. Two of these properties are angular velocity and centripetal acceleration. You will be using two different techniques to calculate centripetal acceleration. You will then be asked to compare the two methods. You will also graph the relationship between linear velocity, centripetal acceleration and the radius. Conceptual questions pertaining to your perceptions of speed and acceleration as you are riding the Apollo 2000 are at the end of the laboratory. Apparatus: stopwatch, calculator, inclinometer Procedure: Look for the Thrill U. sign or any position to the right of the entry area of the ride for a good place to stand when taking the following measurements. This will provide a clear point of observation when doing the off ride data taking. 1. Use a stopwatch to measure the time that it takes for the ride to rotate five (5) times when at full speed. t = ____________________ sec 2. Calculate the time that it takes for one rotation. T= t _________________sec 5 3. The angular velocity, , is the angle that is swept out over a period of time of a rotating object. Calculate the angular velocity. Remember one rotation is 2 radians angular distance. = __________________ rad/sec 1 Apollo 2000 4. When the ride is in full operation, the arms are oscillating inward and outward. Use the inclinometer to find the minimum angle and maximum angle of the arms with respect to the vertical (dashed line) as shown below. Hold the inclinometer as shown. max min 4.73 meters 8.4 meters across center Inclinometer Inclinometer min = ___________________degrees max = ___________________degrees NOTE: If you are using a PASCO or CENCO inclinometer, then you need to subtract from 90 to get the desired angle because they are designed to take angle measurements with respect to the horizontal. 5. Examine the diagram above. On it or the rear of this sheet, sketch and apply your trigonometric rules to calculate the radius of motion of the car when it is at its maximum and minimum positions. You will be using your trigonometric skills to do this. Treat the 4.73 meters as the hypotenuse and add the opposite side of the triangle to the radius of the ride center. radius at minimum angle = rmin = ___________________ meters radius at maximum angle = rmax = __________________ meters 6. By finding the radius of curvature at these two locations we can find the linear speed that you are traveling by using the appropriate equation. Calculate the linear velocity at the minimum position and maximum position. This can be found by actual distance (circumference) calculations but you may find your linear/rotational conversion easier by using V= r. linear velocity at minimum radius = vmin = _________________ m/sec linear velocity at maximum radius = vmax = _______________ m/sec 2 Apollo 2000 7. Whenever an object is moving in a curved path, there is acceleration applied to that object toward the center of the curve. That acceleration, which causes an object to follow that curved path, is called centripetal acceleration. You are going to find the centripetal acceleration of yourself caused by the rotational motion of the ride. Calculate the centripetal acceleration using the values of velocity that you just calculated. centripetal acceleration at vmin = ac(min) = __________________ m/sec2 centripetal acceleration at vmax = ac(max) = _________________ m/sec2 8. Complete the table by using the same equations and methods that you used in sections 5 – 7. By completing the table, you will be graphing the relation of velocity to the radius and the centripetal acceleration to the radius. Complete the table: ANGLE (degrees) RADIUS (meters) VELOCITY (m/sec) CENTRIPETAL ACCELERATION (m/sec2) min 1 max 2 3 max 4 max 3 Apollo 2000 9. Make a graph of centripetal acceleration ac, versus radius, r, by using the coordinate axis below. You will have to scale the axis yourself, so do so appropriately. Graph of ac versus r. centripetal acceleration (m/sec2) radius (meters) 10. Make a graph of linear velocity, v, versus radius, r by using the coordinate axis below. You will have to scale the axis yourself, so do so appropriately Graph of v versus r velocity (m/sec) radius (meters) 4 Apollo 2000 Do the graphs in questions 9 and 10 represent a linear, quadratic, or inverse relationship? How can you be sure it is linear? Include any equations that created the data you graphed. Finding maximum and minimum centripetal acceleration by using force vectors 11. Another method of finding centripetal accelerations is by using vectors. Vectors are the arrows shown below which show size and direction of particular values. Calculate the maximum and minimum centripetal accelerations using the following diagram and the values you found for the maximum and minimum angle. As before, the dashed line is vertical! FN Fc Equation to use FW = mg FN cos mg mg , and cos FN sin ma c FN mg sin ma c cos g tan a c centripetal acceleration at min = ac(min) = ___________________ m/sec2 centripetal acceleration at max = ac(max) = ___________________ m/sec2 5 Apollo 2000 12. Collect data from three other lab groups and put the values into the table provided for you. When working on real world data, it is always best to get multiple values for each measurement. This helps find and eliminate random error in lab work. Comparison table: Answers from Procedure #5-7 GROUPS Answers from Procedure #11 CENTRIPETAL ACCELERATION CENTRIPETAL ACCELERATION CENTRIPETAL ACCELERATION CENTRIPETAL ACCELERATION ac(min) (m/sec2) ac(max) (m/sec2) ac(min) (m/sec2) ac(max) (m/sec2) YOUR GROUP 1 2 3 Analyze the accuracy and precision of the data of the four laboratory groups by comparing the groups' data in the column to each other. Next, analyze the accuracy and precision of the data by comparing the four sets of answers collectively in problem #12. 6 Apollo 2000 Concept Questions 1. How do you perceive the speed of the ride when you are swinging outward? 2. How do you perceive the speed of the ride when you are swinging inward? 3. When the car is at its maximum angle, why don't you feel as if you are going to fall out? 4. Does the change in radius have anything to do with the angular velocity? Linear velocity? Centripetal acceleration? 7 The Antique Carrousel The Physics of Just Going in Circles Introduction: Sometime between 1918 and 1925 W. H. Dentzel built a classic carrousel that Dorney Park obtained in 1995. While simple, the carrousel can demonstrate many basic and advanced concepts of circular motion. In this lab, you will progress from simple to more advanced computations on curvilinear measurement. Part One will address basic tangential speed measurements. Part Two will take you through a series of centripetal accelerations and lastly Part Three will have you analyze the system through angular measurements. Apparatus: Stopwatch, calculator, inclinometer Data Table and Measurements: In order to do all parts of this lab, several measurements will be needed. In the blanks below, measure and record the values indicated: 1. Time for one revolution: ____________________ (Pick a point on the carrousel and time 3 complete revolutions. Divide this by 3 for a more accurate single revolution time.) 2. Angle reading for the inclinometer. (Be careful of the zero degree point. Hold your inclinometer vertically against the upright bar on the horse and read values before and after rotation starts. The angle is the change in these values from the rest angle to the angle it reaches when in motion. You must do this as the bar on the horse is not vertical!) Angle 1 change (inner row)________ Angle 2 change (second row)_________ Angle 3 change (third row)________ Angle 4 change (outer row) _________ 8 The Antique Carrousel Part One Basic Rotational Motion The analysis of rotational motion in the basic sense uses the general equation: V=D/T where D is the distance covered by the horse and T is the time to complete one revolution (a period). D is the circumference of the ring where the horse is located and is found by 2 R, where R is the radius of the horse’s ring (row). Find the velocities of the horses from the inner to outer rows: 1. V1= 2* meters / T=_______________________ (5.8 meters is the radius of the inner row of horses) 2. V2= 2*meters / T=_______________________ (this row radius is 6.6 meters) 3. V3= 2* meters / T=_______________________ 4. V4= 2* meters / T=_______________________ 5. As you observe the motion of the horses, which appear to be going the fastest? 6. How does this compare to the calculated values above? 7. As you ride the horses, what factors make you feel like you are moving? 8. Compare these factors with the speeds you found above. 9 The Antique Carrousel Part Two Accelerations As you ride the carrousel, you may notice a different “feel” between the inner and outer horse row. This is due to the different speeds and accelerations you experience. The human body is a good accelerometer. In which direction do you feel you are accelerating and on which horses is this the most noticeable? ______________________ The general equation for centripetal acceleration is a=V2/R. (Note that V and R vary as we go from the inner to outer ring of horses. Find the values for the four horse rows below: a1=_______________ a2=___________________ a2=_______________ a4=___________________ Do these values match what you felt on the ride? _______________________ We can double-check these values by using the inclinometer data. Since the inclinometer you used shows the net angle between the gravitational and centripetal acceleration components, we can show ac =g * tan In the space below, compute the values for the accelerations of the 4 rows of horses using the angles you measured and tangent equation. a1=_______________ a2=___________________ a3=_______________ a4=___________________ On the axes below, graph the values for your accelerations (found above) verses the radius. What relationship is this? Start your graph with zero and scale (R) and (a) carefully. a R 10 The Antique Carrousel Part 3 Angular computations Many people have trouble understanding the rotational components of motion. They are actually simple to do. Consider the following: Which row of horses takes the longest to go around one revolution? OK, a simple question, they all take the same time. While the velocities differ (as seen in Part One), the time and angle they cover are all the same. We call this measure the angular velocity. If we have the period T from the data we took in the beginning: radians)/T (seconds). What is the angular velocity for the carrousel? This value is the same for all horses, but the tangential velocity differs with radius, R. In general, the rotational measure times radius gives the tangential component. We find V=*R. For example V1=5.8 meters. How does the angular acceleration compare to angular velocity? If we start with ac=V2/R, and substitute V=R, we prove ac =*R. 11 The Antique Carrousel As you see, there is a linear relationship between acceleration, a, and radius, R. Graph this relationship below using the value for you found on the previous page. a 0 2 4 6 8 meters How does this graph compare with the data you found in Part Two? Are angular methods easier for some calculations than others? 12 The Ferris Wheel Observations: The Ferris Wheel is a wonderful experience of vertical circular motion. 1. Describe the feelings you would experience as you move around in the circle. Compare what you feel at the top and bottom of the ride; also compare your feelings on the way up and on the way down. Activity 1 Calculating the magnitude of the linear velocity and centripetal acceleration Part (a) Observe the ride and measure the time for a gondola to repeat one full trip around the wheel. The time for one complete rotation is called the Period and indicated by the letter T. Make sure that the ride is in the midst of a full rotation (i.e. it is loaded and will not stop to pick up or discharge passengers), gather data for at least 3 different trials and find the average period. The distance traveled in one rotation is the circumference of the circle (2R). Using the radius indicated, calculate the velocity from: v = 2 R/t Data Chart for Calculating Magnitude of Velocity and Centripetal Acceleration Radius of Wheel = 12.3 m (40 ft) One Rotation Trial 1 Period(s) Trial 2 Period(s) Trial 3 Period(s ) Average Period(s) Velocity (m/s) Acceleration (m/s/s) Circular motion results from an acceleration directed towards the center of the circle (centripetal acceleration). Find the acceleration using: Centripetal acceleration = velocity squared divided by radius or aC = v2/r See Data Chart for Activity 1 13 The Ferris Wheel Activity 2 Determining the forces acting on a rider at key points To find the force required to keep you moving in this circle, according to Newton’s Second Law of Motion, you need to multiply your mass by this acceleration. Centripetal Force = mass times centripetal acceleration or FC = m aC If mass is in kg and acceleration is in m/s/s, then the unit for force is a Newton (symbol N) Data Chart for Finding Centripetal Force Your mass (kg) = _______ Hint: Your Weight (N) = m * 9.8 m/s/s = _______ To find your mass in kg, you may find it useful to know that the weight of a 1 kg mass on earth is approximately 2.2 pounds. velocity (m/s) Radius of Wheel (m) 12.3 Acceleration (m/s/s) Centripetal Force (N) 14 The Ferris Wheel Part (b) The centripetal force will be the same value throughout the ride. However, the forces that combine to create the centripetal force change as the position on the circle changes. At all positions on the ride the forces add to give a total force towards the center of the wheel. Seat Force Fs A Seat Force Fs Weight SeatBack Force FB Weight Fg Seat Back Force FB Seat Force Fs D B Fg Seat Force Fs Weight Fg C Weight Fg Position A - The seat force and the weight are in opposite directions. The weight must be larger than the seat force to give a total downward force. FS = FW - FC Position B - The vertical seat force and the weight are in opposite directions and are of the same magnitude since the total must add to a force in the direction toward the center of the circle. This force acts on the rider through friction with the seat or through the back of the seat. FS = FW 15 The Ferris Wheel Position C - The seat force and weight are in opposite directions. The seat force must be larger than the weight to give a total force that is upward. FS = FW + FC Position D - The vertical seat force and the weight are in opposite directions and are of the same magnitude since the total must add to a force in the direction toward the center of the circle. This force acts on the rider through friction with the seat or through the back of the seat. FS = FW Using the data calculated in previous activities, find the magnitude of the vertical seat force at each of the 4 locations. If mass is in kg and acceleration is in m/s/s, then the unit for force is a Newton (symbol N). Data Chart for Finding Seat Force Position Fw (N) Your Weight Vertical F C (N) From Activity 2a Fs (N) Seat Force Fw (N) Your Weight Vertical F C (N) Fs (N) Seat Force A C Position B D 0 0 Part (c) Force factors give an indication of what the rider experiences on the ride. In a vertical circle, the force factor (FF) is defined as the ratio of the forces you feel to the force of your weight: Force Factor = Seat Force/Weight or FF = Fs/Fw The resulting number is often referred to as a “g” force, indicating how the force you feel compares to your weight. One “g” means that the forces you feel match your weight. This is what you normally experience. Two g’s mean that the force you feel is twice your weight and many people would indicate that they feel “heavier.” Use the data from Activity 2b and predict the “g” forces acting on you through the four curves: 16 The Ferris Wheel Activity 2c Data Chart for Predicting Force Factors Location Fs (N) Seat Force from 2b Fw (N) Your Weight Force Factor A B C D Activity 3 Measuring “g’s” Someone in your group needs to ride the Ferris Wheel. Using your vertical accelerometer (long tube), measure the g's at the four locations being studied. If possible, take three runs so that you can average your data. Remember that 1 g means that you feel forces equal to your weight, 2 g’s mean that you feel forces that are double your weight, etc. To measure “g” forces, hold the accelerometer parallel to your body (perpendicular to the floor). Carefully observe the accelerometer through one complete rotation and record your best approximation of the reading at the four points of interest. Data Chart for Measurement of “g” Forces Location A Location B Location C Location D Trial 1 g force Trial 1 g force Trial 1 g force Trial 1 g force Trial 2 g force Trial 3 g force Trial 2 g force Average “g” force Trial 3 g force Trial 2 g force Average “g” force Trial 3 g force Trial 2 g force Average “g” force Trial 3 g force Average “g” force 17 The Ferris Wheel Questions for Analysis: 1. Compare your calculated (predicted) force factors with the “g” forces measured on the ride. 2. Where is the “g” force largest? Explain. 3. Where is the “g” force smallest? Explain. 4. Describe what happens to the “g” forces as you complete one full rotation on the Ferris Wheel. 5. Would it be possible to design a Ferris Wheel ride where the passengers feel “weightless” at some point of the ride? Explain your reasoning. 6. Explain the effects of changing the radius of the Ferris Wheel while keeping the speed of the ride the same. Describe the effects for both a larger and smaller radius. 7. Explain the effects of changing the speed of the Ferris Wheel while keeping the radius of the wheel the same. 18 The Enterprise GOING IN CIRCLES Introduction: The Enterprise is a good ride to experience and measure what people call “g's of force.” What they are actually measuring are the forces a body experiences as compared to the standard contact force of mg, which we experience in equilibrium. When contact forces accelerate a body, it is natural to compare the sensation and value to mg, thus the ratio of the force on a body to mg gives rise to “g's of force.” When standing or sitting with no acceleration, the contact force on our body = mg, and we experience a “g” value of mg/mg = 1. Objective: In this lab you will compare calculated "g" values of force of your experience with force meter values as measured on the Enterprise ride. Procedure Part I: Theoretical values During this ride you will be able to experience and measure “g” values for three different situations: moving in a horizontal circle, at the top of a vertical circle, and at the bottom of a vertical circle. Place all responses on the data/calculation tables that can be found within the laboratory. Using the diagrams below, write the equation for the net force on the rider. In the first two cases, it is the net vertical force, in the third case it is the horizontal force. Based on the diagrams, fill in the blanks on the data table, and then solve for the contact force Fs. 19 The Enterprise Part I: Theoretical Results Fill in the blanks based on the diagrams. Diagram A Diagram B Diagram C Fnet = ______ - mg Fnet = Fs + ______ Fnet = ______ Newton's Second Law says Fnet = _______ Finally, solving for the contact force Fs Bottom Top Horizontal Fs = ma + ___ Fs= ____ - mg Fs= ____ Part II: The Experience To do this part you must go on the ride. When you are on the ride, sit on your hands, if possible, so you can better feel the force of the seat on your body. It may also be beneficial to shut your eyes at the key points of the ride so your sensations are not biased. As the ride commences take note of the push of the seat on your body, and try to compare it to the 1g feeling of the seat when you are first strapped in. After the ride is over, record on your data table whether the g's are greater, less than, or equal to 1 at each of the key positions. g values >, <, or = 1 Horizontal Top of vertical Bottom of vertical ____________ ___________ ____________ 20 The Enterprise Part III: Numerical 1. Before going on the ride, you will need the centripetal acceleration of the ride at top speed by using the period and radius. The radius is taken from the blueprints and is 8.5 m. This is on your data sheet. For you to measure the period, you must climb the hill a bit to get a good view of the ride. Choose a rider or car that is easily identified. Wait until the ride is at full speed (you can tell by the sound) and time. (two revolutions). Calculate the period, which is the time for one revolution. Car # 5 is marked with a red spot. 1. r = 8.5 meters Number of revolutions Total time Time for one revolution _________ ________ ____________ 2. 2 2 Calculate the centripetal acceleration using ac = 4 r / T 2 2 ac = 4 r / T ac = _____ / _____ = ____________ number substitutes answer 3. Now you are ready to record the “g” values by taking a force meter on the ride. Record the readings in a horizontal circle, at the top of the loop and at the bottom of the loop. Force meter readings: a. Full speed in a horizontal circle: __________________ b. At the top of the vertical loop: __________________ c. At the bottom of the vertical loop: __________________ 21 The Enterprise 4. Since the force factor Ff is a ratio of Fs to mg, the equations in Part I become: a. Horizontal circle: Ff = Fs/mg = ma/mg = a/g b. Top of loop: Ff = Fs /mg = (ma -mg)/mg = a/g - 1 c. Bottom of loop: Ff = Fs/mg = (ma + mg)/mg = a/g +1 Calculate the predicted force factor value for each situation. The value of (a) is the centripetal acceleration and (g) is 9.8 m/s2 on earth. Horizontal circle : Ff = Fs/mg = ma/mg = a/g = ______/_______ = _______ Top of loop: Ff = Fs /mg = (ma -mg )/mg = a/g – 1 = ______/______ -1 = _______ Bottom of loop: Ff = Fs/mg = (ma + mg)/mg = a/g +1 = _____/______ + 1 = _______ Conclusion: Do the “g” values recorded compare reasonably well with those calculated from the centripetal acceleration? Support your answer. Conclusion: ______________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ 22 Revolution The Revolution is a unique and exciting ride that combines two of the most frequently discussed motions in physics, pendulum and rotational. As the ride picks up speed the passengers are set into these two motions simultaneously, producing an unusual sensation of motion not experience in daily life. As you watch this ride you will take time measurements involving both the pendulum and rotational motions then do some calculations to determine the amount of force acting on the riders and compare the pendulum motion with that of a simple pendulum. Revolution The ride consists of a large, vertical beam that is swung back and forth like a simple pendulum. At the bottom of the beam, a large circular ring type of arrangement is attached. The riders sit along the outer rim of the ring. As the pendulum swings, the ring rotates. You will measure the period of oscillation of the vertical beam and the period of the circular motion of the ring. 23 Revolution DATA: Length of the vertical beam: L= 7.25 m. (added dimension is 10.24 meters) Radius of the ring: r= 4.26 meters The Theory A. The Pendulum As you may recall from your physics class, a simple pendulum consist of a mass hanging at the end of along string or rod. The period of the motion is the time required to swing through one complete cycle from point A to point B and back to A. See the diagram below. A B The period T is given by T=2 Where L is the length of the pendulum and ‘g’ is the acceleration due to gravity. G = 9.90 m/s2 B. Centripetal Force 24 Revolution Whenever an object of mass ‘m’ is moving on a circular path of radius ‘r’ with a speed ‘v’ there is a centripetal force ‘F’ acting on the object and it is given by the formula: F =mv2 / r This force is always directed toward the center of the circle. The passengers on the ride are seated around the circumference of a circle. As the ring rotates, the passengers feel a centripetal force given by equation number 2. V= Where T is the period (time to complete one cycle). The Procedure A. Pendulum Motion 1. When the ride begins, use a stopwatch to measure the time it takes for the system to complete 4 cycles of the pendulum type motion. From this time, calculate the time for one cycle (the period T). Time for 4 cycles = ________________________________seconds T = ____________________________________________seconds 2. Solve equation number 1 for the length L of a simple pendulum that has the same period as the revolution pendulum. Compare this length to the length of the actual vertical beam. B. Rotational Motion 25 Revolution 3. When the ride begins to move, measure the time it takes for the passengers to complete 4 rotations. From this time, calculate the time required for one rotation. From this time and equation number 3, calculate the speed ‘v’ of the riders. Using equation number 2 to calculate the force on a rider (use your own mass ‘m’) Time for 4 cycles = __________________________seconds T = _______________________________________seconds V = ______________________________________m/s F = ______________________________________N 4. From the force ‘N’, calculate the ratio of the force ‘F’ to the weigh ‘mg’. This would be the so-called ‘g’ force acting on the rider. F /mg = ________________________________ 26 If there was one scientist who would love modern amusement parks, it would probably be Galileo Galilei. The freefall condition he studied so carefully richly experience in this modern day environment. Advanced technology had made experiencing freefall not only safe, but exciting as well. We will analyze freefall in this laboratory activity. Dominator at Dorney Park is two different rides built on one common tower. One side launches you upward at 22 m/s and allows you almost four seconds of freefall condition while you decelerate and return to the launch point. The other side lifts you to 52 meters and launches you downward at nearly 18 m/s where YOU bounce on an air cushion to almost half of the initial height. The two different versions of the ride will be used to analyze two different aspects of motion. Launch side will look at basic kinematics and accelerated motion while the Drop side will examine momentum/impulse and work/energy conditions. Sketch Dominator and mark in relative positions and time data. Launch Side and Kinematics: To study the basic kinematics of Dominator, you will need to observe the following details: Time of launch acceleration_______ seconds (observe the bodies of riders to see when acceleration begins and ends) Time of free fall condition _______________ seconds (observe between acceleration and deceleration periods) Time of deceleration ____________________ seconds (watch when arms and legs drop) When you ride, you may also take acceleration data with your force meter and log them below: launch acceleration force reading ____________ free fall acceleration force reading ____________ deceleration force reading __________________ With the motion data collected, you will be able to find the following values: Since in a frictionless environment (which we will assume since air drag is minimal) 27 Vup=-Vdown V = Vdown - Vup = 2 Vup = g Tfreefall 28 Since we are on Earth, g = 9.8 m/s/s downward and the delta V value is negative (your instructor may ask you to show that!), we can find the Velocity by using the previous equation. Vup = g * _______________=______________ m/sec With this value, we can find the acceleration and deceleration you experience at the start and end of the ride. Note you may get a 4g reading, but the calculations below will be much less. Don’t worry; trust the numbers, differences will be discussed in question #3. Aup= Vup / Tup= _____________m/s/s / _______________sec.= ________________m/s/s Adown=Vdown/Tdown=Vup/Tup=______________m/s / ____________sec.=__________m/s/s Convert these two accelerations to g’s or Force Factor readings by dividing by 9.8 m/s/s. FFup=Aup/9.8m/s/s=_____________ “g’s”, FFdown=Adown/9.8m/s/s=___________”g’s” Questions 1. How do the force factors or “g” readings compare? What are sources of error? 2. What is wrong with the advertising statement? Riders reach speeds of nearly 50 mph almost instantly after takeoff then experience negative gravity before they plummet back towards earth. 3. The specifications for Dominator are a 4.g launch and landing. You probably noticed the 4g reading but did not find 4g’s in your calculations, why? (Hint: average vs. constant accelerations) 29 4. How do the average and maximum values compare? Did you observe linear or nonlinear accelerations? If they are linear with zero as one end point you can use the numeric average. Did that work here? 30 Dominator Part II, The Drop Side: or Work/Energy on the bounce…… By using the inclinometer and standing at some convenient distance from the base of Dominator, find the angle of incline for the maximum height and height after the first bounce. Angle at maximum height ____________________________(note this is 52 meters of altitude) Angle where riders begin deceleration=____________________________ Angle at max height after first bounce ___________________________ By using trigonometry, calculate the height where deceleration begins and after first bounce. The Law of Sine’s works well or use a scaled drawing and find the distance from the ride to your measuring location. Height where deceleration begins=__________________ Height after first bounce ______________________ First measure a total time for the drop, then measure the time the riders are decelerating by watching their bodies. Arms and legs are a good cue to see when deceleration is occurring. T(drop)=_____________ seconds T(decel)= ____________seconds Calculate your energy at these two positions. The maximum Mechanical Energy is the sum of the potential energy at the top and kinetic energy gained during the launch. The ride applies g/2 acceleration for approximately 10 meters. The remainder is covered in the PE calculation. ME(max) = PE + F * d = mgh + mg/2*10m = ___________________(units also) PE (first bounce) = m g hbounce=____________________________________(units also) 31 1. How much energy was lost from maximum height to first bounce height? 2. What is the efficiency of the pneumatic spring used to bounce you? (remember it should not be very elastic, they want you to stop eventually) 3. Did you notice your fall was not “free fall” soon after launch? The pneumatic system begins your deceleration soon after launch. If it were not for that, what would your final speed be before the 10-meter main deceleration? Use KE=PE + Work to find your velocity. 32 4. Compare this velocity with an approximation based on landing time and average deceleration of 2g’s. How do they compare? 5. Since we know the distance the riders came to rest in, using the fact that change in energy is due to work done we can find the average force on you. Find F given F=Energy / distance (work/energy theorem). 6. Compare this to the force found by change in momentum divided by time. (application of impulse/momentum calculations) 7. In all cases above, we have used average values. Calculus students should use the linear change in acceleration and reanalyze questions 5 and 6 using the appropriate F/t and F/d graphs in the area below. This question is meant to be open ended. It is safe to assume the force varies linearly with distance increasing from 0 to the 4g force. Be careful when looking at the time values due to the distance/velocity relationship. 33 The Sea Dragon Sea Dragon The following two activities involve the use of the conservation of mechanical energy and Newton's Second Law to determine the maximum speed of the Sea Dragon. It is recommended that the student first observe the motion of the ride to determine when and where the ride undergoes the motion of a physical pendulum. The portion of the ride studied should be while the boat is traveling freely. Activity I - Maximum Speed Using Energy Objective: To determine the maximum speed of the Sea Dragon using the principle of conservation of mechanical energy. Procedure: 1. Use the inclinometer to measure the maximum angle, , that the ride makes with the vertical. Maximum angle, _________________ 34 The Sea Dragon (Part 1) 2. The length of the swing arm, L is 10.7 m. (See diagram above.) Knowing this and that the maximum angle determines the maximum height, h, use the following guide to find h. h = 10.7(l - cos) meters Maximum height, h_________________ 3. Assuming that mechanical energy is conserved, the potential energy at the maximum height is equal to the kinetic energy, KE, at the lowest point. The lowest point is where the boat is traveling freely with maximum speed, v. PE = KE mgh = 1/2(m)v2 Solve mgh = ½ (m)v2 for v and then calculate the maximum speed, v. 35 The Sea Dragon Maximum speed, v_________________ (Part 1) Activity 2 – Maximum Speed Using Newton’s Second Law Objective: To determine the maximum speed of the Sea Dragon using Newton's Second Law and to compare this value to the speed found in Activity 1. 1. Ride the Sea Dragon. Using a hand held vertical accelerometer measure the "g's" at the lowest point of the ride's swing. Remember that 1 g means you feel the seat exerting a force, FN, on you that is equal to your normal weight, making you feel you normal. Two "g's" mean that you feel like you weigh is twice your normal weight in that the seat exerts a force, FN, on you equal to twice your weight. Number of g’s at the lowest point: _________________ 2. Since the motion of the ride near the bottom of the swing is approximately uniform circular motion, Newton’s Second Law predicts that FN -Fg = (mv2)/r where, FN is your support force (the force that the seat exerts on you) Fg is your weight m is your mass v is your speed and is a maximum value r is the radius of curvature (10.7 m) Solve this equation for maximum speed, v. What is the maximum speed equation, solved for v =_________________ 3. Determine your support force by multiplying the number of “g’s” by your weight. What is the support force, FN_________________ 36 The Sea Dragon (Part 1) 4. Determine your mass using Newton’s Second Law. m= Fg/g Where g is the acceleration due to gravity What is your mass, m_________________ 5. Calculate the maximum speed, v, using the equation in #2 of this Activity. What is your maximum speed, v_________________ 6. Compare the two speeds using a percentage difference. If the speeds do not agree, discuss possible sources of error. What is the comparison of two speeds_________________ 37 The Sea Dragon (Part 2) The following activity involves the use of oscillatory motion concepts. It is recommended that the student first observe the motion of the ride to determine when and where the ride undergoes the motion of a physical pendulum. The portion of the ride studied should be while the boat is traveling freely. Objective: To determine the period of oscillation of the Sea Dragon in two different ways. Procedure: 1. Using a stopwatch, measure the period of oscillation of the Sea Dragon. 1. period of oscillation, T (first way)_________________ 2. Assume that the ride behaves like a simple pendulum and calculate the period using the following equation: T =2(L/g)1/2 where, T is the period of oscillation L is the length of the pendulum (10.7 m) g is the acceleration due to gravity 2. period of oscillation, T (second way)_________________ 3. Compare the two periods using percentage difference. Is the Sea Dragon a simple pendulum? 38 White Water Landing Observations: White Water Landing will give you an opportunity to use the concepts of momentum and impulse to determine the forces acting on you during the “splashdown.” 1a. After observing a number of boats “splashing down,” does the size of the splash vary or is it fairly constant? _____________________________ 1b. If it varies, what observable factors seem to influence the size of the splash? 2a. Is there any time during the ride that riders appear to lunge forward? 2b. If yes, where and why does this occur? Activity 1 Determining the magnitude of the velocity of the boat before and after “splashdown" Potential Energy PE = mgh (Top of the incline) Kinetic Energy KE = 1/2 mv2 (Bottom of the incline) gravitational field g = 9.8 m/s/s height of incline 25 meters Potential Energy Joules (J) Length of Boat 5.2 m mass (x) gravitational field (x) height 1/2 mass (x) velocity squared at Dorney Park Part (a) Velocity immediately before splashdown 39 White Water Landing To find the approximate velocity of the boat immediately before splashdown, we can make the assumption that the Potential Energy of the boat at the top of the incline is completely converted to Kinetic Energy at the bottom. Find the Potential Energy of a passenger at the top of the incline: PE=___________ Lets assume that all of the Potential Energy becomes Kinetic Energy at the bottom of the incline. Use the information already obtained to find the velocity at the bottom of the incline. Part (a): Before splashdown your mass (kg) = _______________ Hint: To find your mass in kg, you may find it useful to know that the weight of a 1 kg mass on earth is approximately 2.2 pounds. height of incline gravitational field Potential Energy at top of hill Kinetic Energy at bottom of hill 25 meters 9.8 m/s/s (J) =_______________ (J) =_______________ Velocity at bottom of the incline (m/s) = _______________ Part (b): After splashdown Measure the time for the complete boat to pass under the bridge (after it has completed its “splashdown”). Observe at least three boats and find the average time for a boat to pass under the bridge. Use the information concerning the length of a boat to find the average final velocity of the boat. Data Chart for Finding Velocity of the Boat Before and After “Splashdown” Trial 1 Time (s) Trial 2 Time (s) Trial 3 Time (s ) Average Time (s) Time to pass under the bridge Velocity after splashdown (m/s) 40 White Water Landing Activity 2 Determining momentum change and impulse acting during the “splashdown” Momentum is defined as the mass of an object times its velocity. Physicists represent the quantity of momentum with the letter p. momentum = mass x velocity or p = mv Use your own mass to determine the momentum of a passenger riding in a boat both before and after splashdown. As a result of the boat splashing down, the momentum of each passenger changes. Find the change in momentum of the abovementioned passenger. momentum change = momentum after splashdown - momentum before splashdown or p = pafter - pbefore (the symbol delta means change) Data Chart for Finding Momentum Changes pbefore (kg m/s) pafter (kg m/s) p (kg m/s) Momentum of an object is changed by the application of an impulse. Impulse is defined as the product of an applied force and the time that the force acts: Impulse = Force x Time for force to act or J = F t The impulse applied to the passenger is equal to the momentum change for the passenger. J = p 41 White Water Landing The time that the force acts to change the momentum is approximately the same as the time that the “splash” lasts, since the splash is a result of the water applying an impulse to the boat and the boat applying an impulse on the water. Observe at least three “splashdowns” and time how long each splash lasts to find an average splash time. From this information, determine the size of the force required to change the momentum of a passenger with your mass. If mass is in kg and acceleration is in m/s/s, then the unit for force is a newton (symbol N). Data Chart for Finding Forces Acting Trial 1 Splash time (s) Trial 2 Splash time (s) Trial 3 Splash time (s) Average Splash time (s) p (kg m/s) Impulse (kg m/s) FA = Average Applied Force - FA(N) Activity 3 Comparing Forces You can now determine how the force applied to the rider to slow down compares with other forces. A common force with which to compare is your weight. Determine how the force applied compares to your weight by using the following: Force Factor = Applied Force/Weight Data Chart If mass is in kg and acceleration is in m/s/s, then the unit for force is a newton (symbol N). FA (N) Applied Force from Activity 2 Fw (N) Your Weight = mass * 9.8 m/s/s Force Factor 42 White Water Landing Questions for Analysis: 1. Compare your force factor with other students’ of different mass. Explain your observations. 2. Predict the size of the force acting on the entire loaded boat (the boat has a mass of approximately 1000 kg when empty). Estimate the total mass of riders and boat. 43 The Scrambler The Scrambler Introduction: The Scrambler consists of two sets of arms, the upper sweep arms and the lower arms, that have different radii and revolve around different points of rotation to produce varying forces. You will be studying the paths of these arms and the cars attached to them, along with their properties, such as angular velocity, tangential velocity, and centripetal force. Apparatus: Stopwatch, calculator Procedure Part I: Stand at some point around The Scrambler so that you can see the entire ride. Watch the ride rotate several times. 1. What direction do the sweep arms appear to be rotating? (clockwise or counter-clockwise) _____________________ 2. What direction do the lower arms appear to be rotating? (clockwise or counter-clockwise) _____________________ Do you notice anything about the ride that seems to be cyclical? Try focusing on one particular car or one spot along the outside of the ride. You may notice that each sweep arm returns to the same point along the outside of the rides’ path every revolution, but each lower arm (and attached car) does not. Before the ride starts, find a car that is closest to the fence surrounding The Scrambler. It should be pointing almost directly at you. Remember this car, as you will be following its motion. Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute e 44 Pag The Scrambler 3. As the ride is rotating, what do you notice about the group of 4 cars to which the car you picked out belongs? 4. How many revolutions does it take for that car to return to the same point that it started at? ____________________ 5. What is the length of the sweep arms and what is the circumference of their path? Length: _________________ Circumference: ____________________ 6. What is the length of the lower arms and what is the circumference of their path? Length: _________________ Circumference: ____________________ Part II: Now you will need your stopwatch and calculator. When the ride is up to full speed, record the time it takes for the ride to rotate 3 times. The easiest way to do this is record the time it takes for one of the sweep arms to pass you 3 times. Then divide by 3 to calculate the time of one revolution. Time of 3 revolutions: _______________________ Period (time of 1 revolution): _______________________ Now calculate the angular velocity, keeping in mind that each revolution is 2*pi radians and your period has the units seconds/revolution. Angular velocity (in radians/second): ________________________________ Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute e 45 Pag The Scrambler The tangential velocity is simply the speed that the object is travelling in its circular path. This can be obtained by multiplying the angular velocity by the radius of the path, or the length of the sweep arms. Tangential velocity (in meters/second): _______________________________ The next part is a little trickier. Your task is to calculate the time it takes for the group of 4 cars to rotate one revolution. The easiest way is to orient yourself to the ride the same way you were before, lined up with the closest car to the outside, and to time 4 rotations of the upper arm, while counting how many times the group of 4 cars spins in a full circle. (Remember: the car completes one spin every time it is swung to the outside of the fence, at the point furthest away from the middle of the ride) Time of 4 revolutions: ______________________ Number of spins in 4 revolutions: _______________________ Period (time of 1 spin): _____________________ You can now calculate the angular and tangential velocities for the lower arms attached to the cars using the same method you used for the sweep arms. Angular velocity (radians/second): _________________________ Tangential velocity (meters/second): _________________________ Part III: Now that you have calculated the velocities of both sets of arms it’s time to use them to reveal some interesting things about The Scrambler. Do the cars on The Scrambler have a higher speed when they are closer to the center of the ride or when they are closer to the outside of the ride? Hopefully, your answer was they travel faster when they are closer to the inside of the ride. Why is this? The speed of the cars is a product of the tangential velocity of both sets of arms. The sweep arms are moving clockwise, while the lower set of arms are moving in a counter-clockwise motion. This causes the tangential velocities of each set Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute e 46 Pag The Scrambler of arms to work together when the cars are closer to the center of the ride and offset each other when the cars are at their furthermost point from the center. Given the information provided above you should now be able to calculate the maximum and minimum speeds of the cars when the ride is at full speed. An important thing to remember when doing these calculations is that while the angular velocity is always constant, the tangential velocity varies as a function of the radius, or the distance of the object from the point of rotation. Also, remember that when the tangential velocities are opposing each other you need to make one positive and one negative. Radius of the lower arm at the point of maximum speed: __________________ Radius of the sweep arm at the point of maximum speed (distance of the car from the point of rotation of the sweep arm): _____________________ Tangential velocity of the car due to the lower arm at the point of maximum speed: _______________________ Tangential velocity of the car due to the sweep arm at the point of maximum speed: _______________________ Maximum speed of the car: _______________________ Radius of the lower arm at the point of minimum speed: __________________ Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute e 47 Pag The Scrambler Radius of the sweep arm at the point of minimum speed (distance of the car from the point of rotation of the sweep arm): _____________________ Tangential velocity of the car due to the lower arm at the point of minimum speed: _______________________ Tangential velocity of the car due to the sweep arm at the point of minimum speed: _______________________ Minimum speed of the car: ___________________ (does this answer surprise you?) Part IV: The final part of this exercise is to calculate the centripetal force on the rider at the innermost and outermost points on the ride’s path. This must be done similarly to the way you calculated the tangential velocity of the car at both points, as centripetal force is also a radius dependant value. The centripetal force always points toward the center of rotation for each set of arms, so be sure to make one positive and one negative, if needed. The equation for centripetal force is Force = mass * radius * (angular velocity)2. You can use your own mass (in kg) in this calculation. (1 kg = 2.2 lbs.) Centripetal force on the rider at the innermost part: _______________________ Centripetal force on the rider at the outermost part: _______________________ Length of Sweep Arm: 4.23 meters Length of Lower Arm: 3.65 meters Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute e 48 Pag The Scrambler Pivot arm from pivot to car: 1.82 meters Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute e 49 Pag The Wave Swinger A SWINGING TIME Introduction: The Wave Swinger is a fairly simple ride, but it does have some interesting aspects. The rate of the rotation of the ride is constant and there are only two forces acting on the swing itself. This allows the easy analysis used in Part IV. However, because the ride tilts, the plane of swing is not horizontal. This adds some interesting possibilities to the motion of the swing and rider. Part I Theory Shown below is a crude picture of a rider on the swing seat. The swing seat and rider will be treated as one object. Since the forces F1 and F2 (actually there are four) are in the same direction, they can be considered as a single force F1 and F2 = F. This is shown below. Procedure: Place all answers where indicated within the procedures. 1. Is the force of the chain F or mg? ________________________ 50 The Wave Swinger 2. The force mg is the force due to what phenomena? ____________________________________________________ 3. Draw the diagonal across the parallelogram. 4. The diagonal of the parallelogram is the magnitude of what force? ____________________________________________________ ____________________________________________________ 5. The direction of the force in number 4 is the same as the: speed or acceleration of the object? ____________________________________________________ 6. If the speed is constant, which acceleration is equal to (0), the tangential acceleration or radial acceleration? ______________________________________________________________ 7. Is v2/r the tangential or radial acceleration? Note: radial is another name for centripetal. ______________________________________________________________ 8. Noting that the diagram in #3 shows the net force (thus acceleration) to be slightly up, would you conclude that the swing is at a low or high spot in its rotational path? 51 The Wave Swinger Part II Observations relating to the theory Procedure: Make the following observations of the ride. 1. Using the outside swings only, does the weight on the swing effect the angle of the chains? ____________________________________________________ Note: comparing an empty swing to a loaded one can do this. 2. Do the riders on the inside swings travel at a faster or slower speed than those riding on the outside swings? ____________________________________________________ 3. Use the inclinometer (as shown in the diagram below) to measure the angle that the chains make with the vertical for when the swings are at their highest point and lowest point. Within experimental error, are they the same or different? ____________________________________________________ 4. Which swings have the greater angle from the vertical? The outside swings or the inside swings? ____________________________________________________ 52 The Wave Swinger Part III Observations to be made on the ride 1. Watch the chain as the ride begins. Which way does it move relative to you? ____________________________________________________ 2. As you ride, how does the force of the seat on your “bottom” feel at a low point as opposed to that at the high point in your rotational path? Note: This force variation is indicative of the force variation in the chain. ____________________________________________________ Part IV Numerical Analysis Objective: To determine whether the radial (centripetal) acceleration is equal to the acceleration value of g (tan ) where is the angle that the chains make with the vertical. Procedure: 1. Use the inclinometer to measure the angle that the outer swings make with the vertical as shown in the figure of Part II, Procedure step 3. If you feel there is significant difference between the angle when the swings are at a low point and high point, record both angles. Angle Value (low point) ________________________________________________ Angle Value (high point)________________________________________________ 2. Time the ride for five complete revolutions. ________________________ *The radius used is given as 9.0 meters. 53 The Wave Swinger Analysis: 1. Calculate the predicted acceleration using g(tan ), where g= 9.8 m/s2. If you used two different angles do this, perform calculation twice. ____________________________________________________ 2. Find the period of the swing. (This is the time for one revolution.) ____________________________________________________ 3. Calculate the acceleration using a = 42r/T2. ____________________________________________________ 4. Compare the accelerations in Parts 1 and 3 using percent difference. ____________________________________________________ 5. Bonus: Based on the force diagram and a vertical component of the chain force equal to mg, show how Newton's Second Law produces an acceleration of g(tan). 54 The Physics of Since the Gravity Rides of the 1500’s, the concept of the roller coaster has been a thrilling challenge for both rider and engineer. In this lab, you will have the chance to test the design of Steel Force, the “longest, tallest, fastest coaster in the East.” This laboratory is divided into 3 sections. Each section is a necessary step to evaluate the next section, so work in order and go as far as your instructor requires. The sections will help you examine kinematics, work/energy theorem, and curvilinear motion. The diagram below shows the parts of the ride we will analyze. Data Point A: is anywhere about half way up the first hill. At this point, the chain drive system is pulling the train uphill at a constant velocity. Data Point B: is at the bottom of the first hill. At this point, there is a tunnel 34 meters long. You will use the length of the tunnel to find the speed at this point on the ride. Data Point C: is at the top of the second hill, 49.1 meters above the ground. You will use the length of the train to find its speed here. Data Point D: is the spiral curve at the far end of the coaster. You will measure the time to go around the curve for one full revolution. The radius of this turn is 31 m. Data Point E: is the return camelback’s first “hump.” You will use the train length to find its speed at this point. All data can be logged on the Steel Force Data Sheet, which follows this lab. The data will help you complete the computations for all of the following sections. 55 Steel Force Section #1 Koaster Kinematics...... One of the primary measurements we must take in physics is the motion value called speed. In this section, we will compute the speed and acceleration values at the five points of interest on Steel Force. In general, we will use: v =d /t and a = v /t. Speed at points A, C, and E: At these points you will compute speed by using train length divided by time taken to pass a point. The train is 19.6 meters long. Check your Steel Force Data Sheet for the time to pass a point on the track at each of these locations. Use the measured times and complete the calculations in the chart on the next page for each of the points A, C and E. Speed at point B: This is the tough one! The tunnel is 34 meters long. If you measure the time through the tunnel, it will be short and you can compute the speed by the basic equation in chart line #4. Can you think of a way of measuring this more accurately? If you can, take the measurement your way and the way described above, then compare. If not, see if you can find another group who has done this. Hint: the train is 19.6 meters long and its length will increase the time of passing a point. Speed at curve D: Your measure of the time around the curve, along with the distance traveled, will give you this solution in line #5. Acceleration on first hill: Now apply your acceleration equation to solve for the average acceleration on the first hill. This is done in line #6. You will need the velocity at the top and bottom of the hill (Data points A and B) and also the time down the hill. Measure the time from when the center of the train passes the top of the hill to when that point enters the tunnel. 56 Steel Force Deceleration on second hill: Using the same type of calculation, find the deceleration rate when going up to point B in line #7. Line 1 VA=19.6m/____________ seconds = ______________ don't forget units VC=19.6m/____________ seconds = ______________ units VE=19.6m/____________ seconds = ______________ units VB=34 m/____________ seconds = ______________ units VD=*62 m/_________ seconds = ______________ units Ahill=V/t =(________-_________)/____________ = ______________ units Ahill=V/t =(________-_________)/____________ = ______________ units Line 2 Line 3 Line 4 Line 5 Line 6 Line 7 Section #2 Using the Work/Energy Theorem...... Once you have been lifted to the top of the first hill, your trip is entirely controlled by a simple concept in physics, work/energy. The motor drive system simply is designed to propel you to the top of the first hill. This motor/chain drive acts on you and the train but, for this example, we will only work with you (all other parts like the train and passengers are proportionately larger). We will work on the assumption that friction is negligible to make our calculations easier. 57 Steel Force 58 Steel Force Up the first hill: As you are pulled up the hill, the motor system must apply a force parallel to the hill in order to move you along. This hill is at an angle of 25 degrees. 8. Find the force on you as you go up the hill: Fparallel =__________________ Note: Recall the equation F par = mg sin This force will be applied up the entire length of the hill, 144 meters. 9. What work is done on you during this part of the trip? Work = ___________________ Before going further, how much power is required to pull you up, if you reach the top in the amount of time that you found? Find this in both Watts and Horsepower. P = #9_______/_______ (time) = _____________watts = _____________hp (10) Now, back to Work... The work done on you in a frictionless environment would remain as part of the total mechanical energy. 11. At the top of the hill, what two types of energy do you have? 12. List them and compute their values for you below. Use your mass for this analysis: (11) E (total mechanical energy) = _______________+________________ name of one (12) E (total mechanical energy) name of the other = _____________+____________ = compute P compute K Since this value will remain nearly constant between the first and second hills, find the values for the stored part of the energy at points B and C. Note the first hill (at B) is below the starting level by 1.5 meters. P at B = _____________ (14) P at C = _____________ (13) 59 Steel Force The work done to get you to the top of the 61 meter hill can be found the same way as finding the stored energy. 15. Using this type of calculation, what is the stored energy at the top of the hill? P at Top = _______________ Does this compare favorably to your calculation from calculation #9? Why? _____________________________________________________________ _____________________________________________________________ _____________________________________________________________ If we assume Einitial=Efinal, we can use the equation: _______________ (value from 12) = P + K. With this we can solve for velocity at two points of interest, points B and C. E(12) = P + K therefore K = E(12) - P Find the values for the motion energy at B and C; then compute the velocity from the equation K =1/2 m v2. K B = _______________ therefore VB = _________________ K C = _______________ therefore VC = _________________ We have measured the velocity at points B and C. Using Work/Energy we have predicted it. Compare the two values. How do they compare? What sources of error exist and how bad were they? In the blocks below, assuming the actual measures from Section #1 to be accurate, compute experimental error and explain. Data Analysis at Point B Data Analysis at Point C 60 Steel Force Section #3 Curvilinear Motion and Vectors reaction forces The basic motion we experience on roller coasters was explained in the sixteenth century. We will look at these principles using the physics and trigonometry studied in class. Mg Fc When you are traveling in the train, your body is moving in a straight line until the track or gravity changes your motion. We will start by looking at changes in the horizontal motion that your train travels. When you travel through the far point spiral, you are traveling at a speed calculated in equation. You have the radius and speed, so find the centripetal acceleration. acentripetal = V2/R = _______2/31 meters = _________________ The average angle of the track at this spiral is 44o. Compare this to a vector diagram of the track’s gravitational force verses its centripetal force. Fill in the values and draw scaled reaction forces with resultant on the drawing above. What do you notice? Discuss below: (include not just values, but what you felt, which way the forces acted on you, etc.) gravity Points B, C and E are other interesting positions. In these areas, you have a combination of seat force forces acting on you in the vertical directions. If we assume a person to be ideally viewed as shown, create a free body diagram (FBD) including the seat and gravitational forces on the rider. Use the dot to the right for your FBD. 61 Steel Force At point B, we will define the upward force of the seat to be N for normal force. The centripetal force is also upward and gravity is downward. From this, we can predict the seat force on you by applying Newton’s Second Law as follows: F = m a The values, that we can see from the FBD, expand to the following: N - m g = m v2/r, or N = m(g+v2/r) 16. Since these values are all known, we can easily find the force on your body. The radius of the tunnel curve is 34m. Using your value for speed from (4) and your mass, find the seat force N. N = ___________________ 17. Divide this (16) by your weight and compare to your accelerometer reading. N/mg =_____________ g’s compared to ___________________ g’s 62 Data Taking Sheet for Side 1, Kinematics Data Point A, You are looking for the velocity of the train going up the hill. Find the time for the train to pass a point on the hill. Data Point C, The velocity at the top of the second hill is again found using train length and t =________________ Time, t, =_________________ Data Point B, Finding the velocity at the bottom of the big hill. Use the time it takes the train to go through the tunnel. Be careful, it will be a very short interval...... t =____________________ Data Point D, To find the speed through the curve, use the circumference and time to pass a vertical point through one revolution. Data Point E, The top of the first Camelback Hump will be used as a reference. Measure the time it takes the train to pass the very top point. t =_________________ t =_______________ 63 Data Taking Sheet for Side 2, Curvilinear Motion and Vectors Data Point A, As you go up the hill, which way do you feel a force? Using your accelerometer, find the acceleration you are experiencing. a=_______________ Data Point A to B, as you accelerate down the hill, you should see your accelerometer reading change. Find the acceleration going down the hill and at the bottom. adown=__________ abottom=__________ Data Point D, We will need acceleration through this curve. Use your accelerometer to measure this value. Data Point E, What is the acceleration at the peak of this hump? Take a reading and listen to the train on the track. What do you notice? a=________________ a=________________ 64 Energy Curves for Steel Force Objective: To investigate a rider’s energy curves for a portion of the Steel Force ride Equipment: stopwatch, scaled photo of Steel Force’s second hill (see Diagram 1 in data section), small ruler Note: In this investigation the x-axis runs along the track. Procedure: 1. Determine the speed of the coaster at position A (0 meter). To do this, time how long it takes the 19.6 meter long train to pass position A and record. (See Diagram 1 in data section). 2. Repeat step # 1 for each of the remaining positions (B through L). 3. Record your weight in pounds. 4. Using Newton’s Second Law and the fact that 1 pound equals 4.448 Newtons, calculate the rider’s mass in kilograms and record. Show your work in the analysis section. 5. Determine the rider’s kinetic energy at each position and record. 6. Using Diagram 1 with its provided scale (the 20 m width), determine the distance each position is from position A (0 meters) and record. Note: This distance is equal to the magnitude of the position, x. 7. Using Diagram 1 with its provided scale (the 20 m width), determine the height each position is from the ground and record. 8. Determine the potential energy of the rider at each position and record. Show your work in the analysis section. Special thanks to Jeff Wetherhold 61 Energy Curves for Steel Force Data: E F height of hill = 46 G H D m I C J B K A, 0 m L 20 m Diagram 1 Length of train = 19.6 meters Rider’s weight = _____ pounds Rider’s mass = ______ kilograms Special thanks to Jeff Wetherhold 62 Energy Curves for Steel Force Position mark A Position,x (m) Position,y (m) Time to pass position mark Speed at position mark (m/s) Kinetic energy at position mark, K (J) Potential energy at position mark, U (J) Total energy at position mark, E (J) 0 B C D E F G H I J K L Special thanks to Jeff Wetherhold 63 Energy Curves for Steel Force Analysis: 1. Show work for finding the rider’s mass. 2. Show work for finding the rider’s kinetic energy. 3. Show work for finding the rider’s potential energy. 4. Using the provided graph paper, graph the kinetic energy, the potential energy, and the total energy as a function of the position, x. Plot the energies on the same set of axes. 5. Based on analysis # 4 results, construct the corresponding net force vs. position graph (use the same piece of graph paper that you used for analysis # 4). 6. According to the net force vs. position graph, what is the net force on the rider at the top of the hill? Does this make sense to you? Explain. 7. Is the mechanical energy of the rider conserved? If not, what happens to the lost mechanical energy? Special thanks to Jeff Wetherhold 64 Centripetal Force and Steel Force Objective: To determine the centripetal force on a person riding Steel Force Equipment: stopwatch, scaled photo of Steel Force’s second hill (see Diagram 1 in data section), small ruler Procedure: 1. Have someone ride the Steel Force and measure, with the vertical accelerometer, the “gforce” at the top of the second hill (see Diagram 1) and record. 2. Time how long it takes the 19.6 meter long train to pass the top and record. 3. Record the rider’s weight in pounds. 4. Using Diagram 1 and its scale, determine the radius of the curvature of the hill at the top and record. Data: TOP 20 m Diagram 1 Special thanks to Jeff Wetherhold 65 Centripetal Force and Steel Force g-force at top of hill = _______ g length of train = 19.6 meters time for train to pass top = ______ seconds rider’s weight, W = ______ pounds radius of curvature of the second hill at the top, r = ______ meters Analysis: 1. Draw a force diagram for the rider at the top of the hill. The forces involved include the normal force, F N and weight, W. 2. Knowing that 1 pound equals 4.448 Newtons, determine the rider’s weight, W in Newtons. 3. Knowing the g-force on the rider, determine the normal force on the rider at the top. For example, if the rider measured 2 g’s, then the normal force on the rider would be equal to two times the rider’s weight. 4. Using Newton’s Second Law and the fact that 1 pound equals 4.448 Newtons, calculate the rider’s mass, m in kilograms. 5. The centripetal force on the rider is equal to the net center directed force, Σ F on the rider. Use this fact to determine the centripetal force, F c on the rider at the top. 6. Knowing the length of the train and the time for the train to pass the top, determine the speed, v of the rider at the top of the hill. 2 7. Knowing that acceleration of the rider at the top is given by the equation ac = v , r determine the rider’s acceleration. 8. From Newton’s Second Law, the centripetal force on the rider is also equal to the rider’s mass times the rider’s acceleration or F = māc . Use this fact to determine the centripetal c force on the rider at the top. 9. Using a % difference, compare the centripetal forces you found in analysis # 5 and # 8. Special thanks to Jeff Wetherhold 66 The Talon will allow the opportunity to study the forces that act as your body goes through a variety of loops and curves. Before riding, spend some time looking at the ride. If possible watch a number of trains going through the complete circuit. 1. Describe what your body would expect to feel at the following points on the ride: (See the accompanying diagram to identify these points) o Bottom of the first hill: o Top of the vertical loop (when you are upside down): o As you pass through the top of the Zero “g” Roll (the title of this element may be helpful!): o The middle of the horizontal Spiral: Formulas required for these activities: Mass in kg = Magnitude of velocity = Centripetal acceleration = Centripetal Force = Force due to gravity (weight) = Weight in pounds/2.2 Distance traveled/time interval Velocity squared divided by radius Mass times centripetal acceleration Mass times gravitational acceleration or or or or Force Factor = Seat Force/Force of Gravity or v = d/t aC=v2/r F = mac Fg=mg (g = 9.8 m/s/s at Dorney Park) FF = Fs/Fg 67 Most of the required measurements can be taken while observing the Talon from the area around the Antique Carousel near the Main Gate. Part 1 - Determining the magnitude of the velocity at key points on the ride. Observe some cars traveling through the ride. Find the magnitude of the velocity of the cars as they pass each of the following locations. To find the velocity, use the length of the train (12.2 meters) and measure the time it takes the complete train to pass a certain point. Be sure to collect data for at least three trials and average your results. Measure time in seconds (s) and calculate the velocity in meters per second (m/s) Bottom of the first hill: Length of Train = 12.2 m Time for train to pass the point = _______________ Magnitude of the velocity of train = ____________________________ Top of the vertical loop (when you are upside down) Length of Train =12.2 m Time for train to pass the point = _______________ Magnitude of the velocity of train = ____________________________ At the peak of the Zero “g” Roll Length of Train = 12.2 m Time for train to pass the point = _______________ Magnitude of the velocity of train = ____________________________ In the middle of the horizontal Spiral Length of Train = 12.2 m Time for train to pass the point = _______________ Magnitude of the velocity of train = ____________________________ 68 Part 2 - Determining the accelerations and forces acting on a rider at key points: You will need to have your mass in kg determined: Your mass = ___________kg The accelerations and forces experienced moving through a curve or loop can be considered by using the principles of circular motion. Bottom of the first hill: Radius of curve= 25.0 m Centripetal Acceleration = ____________ velocity (from part 1) = ______________ Centripetal Force = _________________ Top of the vertical loop (when you are upside down): Radius of curve= 6.0 m velocity (from part 1) = _____________ Centripetal Acceleration = ____________ At the peak of the Zero “g” Roll Radius of curve= 18. 0 m Centripetal Acceleration = ____________ In the middle of the horizontal Spiral Radius of curve= 9.1 m Centripetal Acceleration = ____________ Centripetal Force = ________________ velocity (from part 1) = _____________ Centripetal Force = ________________ velocity (from part 1) = _____________ Centripetal Force = ________________ 69 Part 3: Determining the force that a rider feels at key points and calculating expected “g” forces. In addition to moving along the curve, a force is also required to “hold you up”. This additional force would be an upward force equal in amount to your weight. The centripetal forces that you calculated in Part 2 are simply a combination of the force that the seat exerts (Fs) and the force due to gravity (Fg). The force due to gravity is often referred to as your weight. Calculate your weight in Newtons. Force due to gravity (weight) = ____________________N Bottom of loop. Fc is up, so Fc=Fs-Fg Seat Force Fs = Fc+Fg Fs Top of loop. This situation works for top of vertical loop and top of Zero “g” roll Fc is down, so Fc=Fg+Fs Seat Force Fs = FC-Fg Fs Fg Fg Your weight in newtons = __________________N Bottom of the first hill: Centripetal Force (from part 2)= ____________N Seat Force = ____________________N Top of the vertical loop (when you are upside down): Centripetal Force (from part 2)= ____________N Seat Force = ____________________N 70 At the peak of the Zero “g” Roll Centripetal Force (from part 2)= ____________N Seat Force = ____________________N Analysis of the forces in the horizontal spiral requires knowledge of vector mathematics. This analysis may be optional. The forces in the horizontal spiral are a bit more complicated, the centripetal force is a combination of the force required to hold you up (opposite of force of gravity) and the seat force, both of which are vectors. Since these are vectors that are not parallel to one another you need to use vector addition techniques. Seat Force (Fs) Force holding you up = your weight (Fg) Centripetal Force (Fc) From the Pythagorean Theorem: Fs2 = Fg2 + Fc2 In the middle of the horizontal Spiral Centripetal Force (from part 2)= ____________N Seat Force = ____________________N Part 4: Calculated “g” Forces: Of interest to many roller coaster enthusiasts are the “g” forces experienced at various places on the ride. Use the calculations you have just completed to find the Force Factor (or “g” forces) that you can expect at the key points on the ride. Bottom of the first hill: Seat Force (from part 3)= ____________N Predicted Force Factor (“g” force) = ______________ Top of the vertical loop (when you are upside down): Seat Force (from part 3)= ____________N Predicted Force Factor (“g” force) = ______________ At the peak of the Zero “g” Roll Seat Force (from part 3)= ____________N Predicted Force Factor (“g” force) = ______________ In the middle of the horizontal Spiral Seat Force (from part 3)= ____________N Predicted Force Factor (“g” force) = ______________ 71 Part 5: - Measuring “g’s” Someone in your group needs to ride the roller coaster. Using your vertical accelerometer (long tube) measure the g's at the points being studied. Remember 1 g means that you feel forces equal to your weight, 2 g’s mean that you feel forces that are double your weight, etc. To measure g forces, hold the accelerometer parallel to your body (perpendicular to the lap bar). As you ride, try to remember the readings as you pass through each of the key points, do not attempt to write down the readings in the midst of the ride!!! Bottom of the first hill: Measured “g” Force = ______________ Top of the vertical loop (when you are upside down): Measured “g” Force = ______________ At the peak of the Zero “g” Roll Measured “g” Force = ______________ In the middle of the horizontal Spiral Measured “g” Force = ______________ 72 Questions for Analysis: 1. Which of the four points has the rider traveling at the greatest speed? Explain why this is the fastest of the four points. Dorney Park ads say that Talon reaches speeds of 58 mph, how do your results compare to this claim? (either convert your results to mph or convert 58 mph to m/s to do the comparison) 2. Compare the calculated force factors at each point to the measured force factors. Why may there be some differences? 3. Why do they refer to the third element studied as the Zero “g” Roll? Do your results seem to agree with this claim? 4. Why is the radius of the vertical loop so much smaller at the top than at the bottom? How do you think the experience of Talon would be affected if the vertical loop had a large radius at the top (like it does at the bottom)? 5. Describe what factors make Talon exciting and different from other coasters like Steel Force or Hercules? If you studied another coaster, compare the results and explain what makes the other coaster exciting and different from Talon. 73 Immelman Inclined Spiral Zero “g” Roll Vertical Loop Horizontal Spiral Bottom of First Hill Thunderhawk An Enlightening Lab Introduction: Thunderhawk is the original roller coaster for Dorney Park. Although it looks small compared to Steel Force it is an excellent ride in design and function. As with all wooden type coasters the vibrations are part of the experience. It is for this reason this lab has no measurements taken on the ride though it is highly recommended you ride it to experience the usual thrills and also the decrease in energy as you move from beginning to end. Purpose: To measure the lost mechanical energy from the top of the first hill to the small hump near the end of the ride. Theory: There are no blueprints of this 1923 ride so all measurements must be determined by you. Position yourself in the vicinity of the ride called, Possessed so that you have an unobstructed view of the first hill of the Thunderhawk. In line with the top of the hill and approximately twelve feet off the ground you will see a red spot. Notice that this spot is the same height as the small hump behind the hill. See figure one. This small hump is near the end of the ride. This height will be zero potential energy, thus when the coaster goes over the small hump it will have all kinetic energy and no potential energy. Note: potential energy is based on position relative to a zero reference level. Any height below the red spot would be a negative potential energy and the kinetic energy would be more than our value. Top of hill 6 ft Top of hump Red dot Figure 1 75 Thunderhawk An Enlightening Lab You will find the mechanical energy you have left at the top of the small hump as a percentage of the mechanical energy you have at the top of the first hill. Since energy is conserved this "lost" mechanical energy is actually converted to small molecular motions associated with thermal energy. This percentage is in a sense a measure of our coaster's efficiency. Equations: 1. Top of the first hill: Total Mechanical Energy PE KE mgh 2. 2 2 mv Top of the hump: Total Mechanical Energy KE 3. 1 1 2 mV 2 Fraction of Mechanical Energy Remaining 2 1 mV 2 1 mgh mv2 2 2 1 V 2 (mass cancels ) 1 gh v2 2 4. V 2 2 gh v2 ( multiply by two) Percentage of Mechanical Energy Remaining 2 V 2gh v2 100 76 Thunderhawk An Enlightening Lab Procedure and Data: 1. Using the red spot as h = 0 determine the height of the first hill given that the vertical boards are 6 feet long. Notice we will be using English units, so g = 32 ft/s2. Also the nearest whole foot will be uncertain so generally we will be working with two significant digits. Estimated height of hill as measured from the red spot h = _____ feet 2. The trains are 40 ft long. Measure the time it takes the entire train to pass a vertical rail at the top of the hill. time = _________ seconds 3. Now measure the time it takes an entire train to pass over the small hump behind the first hill. This is near the end of the ride. You can use the top of the hump as the reference point. You can move your position to line up a vertical board with this spot. time = _________ seconds Calculations: 1. Calculate the speed v in ft/s as the train passes over the first hill. 2, Calculate the speed V in ft/s as the train passes over the hump. 3. Use equation 4 from the theory to calculate the percentage of the energy remaining near the end of the ride. 77 Thunderhawk An Enlightening Lab Want an "A”? Answer the following: 1. If the weight of the train is 4000 pounds find the potential energy at the top of the first hill. Note: the unit "pounds" is weight, therefore mg = 4000 pounds so just multiply by h. The unit will be ft-lb. 2. Find the mechanical energy "LOST" during the ride. 3. If this energy were heat what would be the temperature increase of a cup of water (0.52 lbm) if the specific heat of the water were 1 BTU / ( lbm oF). Assume no heat was used to heat the container. 1 BTU = 778 ft lb A BTU is a British Thermal Unit 78 The Hydra Introduction: This experiment is written in three different parts. Lab 1 will be using the speed of the Hydra train at various places on the ride along with the radius of curvature of the track at those locations to calculate the force factor that the rider experiences. Lab 2 will also calculate force factor but this time the track is banked which makes the problem a little more challenging. Lab 3 will be calculating the total amount of energy at various places looking at the amount of energy lost throughout the ride. Preliminary Data: The information found in the Preliminary Data section will be used throughout the Hydra Labs. Equipment needed: Stopwatch Vertical accelerometer Special thanks to Brent Ohl 79 The Hydra D C B G A E F Measure the time it takes the train to pass the reference points shown below. Start the stopwatch when the front of the train reaches the reference point and stop the stopwatch when the back of the train reaches that point. The pictures will help you find the reference points. It is recommended that at least two people measure the time and take the average for more accurate results. Enter the values of time in Table #1. Point A: The bottom of the first hill. Look for the red dot on the middle of the track. Special thanks to Brent Ohl 80 The Hydra Point B: The top of the zero-g roll. Use the track junction as the reference. Point C: The bottom of the hill just after the zero-g roll. Look for the red dot. Point D: The middle of the cobra roll (a.k.a. Happy Face). Use the support post as the reference point. Point E: The top of the camel back just after the train passes the station. Use the support post as the reference point. Point F: The middle of the spiral near the end of the ride. Use the support post as the reference point. Point G: The end of the ride just before entering the breaking segment. Use the first vertical post on the handrail as the reference point. Now ride Hydra and measure the force factor at Points A-F using the vertical accelerometer holding it parallel to your upper body. Once again, it would be best if at least two people measure the force factor and compare them for more accurate results. Enter the values for the force factor in the table below. In the table below use the descriptors; lighter, heavier, or same to describe the sensation you experienced at the designated locations in reference to how you feel motionless, upright. Special thanks to Brent Ohl 81 The Hydra Knowing the that the length of the train is 12.298 meters long, ∆x, and v x , calculate the speed of the t train at the designated locations. Table #1 Track Section Time, ∆t (seconds) Train Speed, v (m/sec) Force factor Sensation (lighter, heavier, same) A B C D E F G XXX XXX This information will be used throughout the three labs for Hydra. Your weight in lbs, Fw = ________________ x 4.45 N Your mass in kg, m = lbs = __________________N Fw ______________________ kg g Special thanks to Brent Ohl 82 The Hydra Lab 1—Force Factor Analysis Using the orientation represented in the picture for each location, draw the force diagram of the rider. Then use Newton’s laws to calculate the force of the seat and force factor experienced by the rider. The radii of curvatures are given for each part. The speed of the train was calculated in table #1. Part A Radius of the track, r = 26.25 m Calculate the seat force using: Fseat Fw force factor mv 2 r Fseat ___________ Fw Part B – top of zero-g roll Radius of the track, r = 16.0 m Calculate the seat force using: Fw Fseat force factor mv 2 r Fseat ___________ Fw Part C – After zero-g Roll Radius of the track, r = 20.8 m Calculate the seat force using: Fseat Fw mv 2 r Special thanks to Brent Ohl 83 The Hydra force factor Fseat ___________ Fw Part D—Cobra Roll (a.k.a. Happy Face) Radius of the track, r = 15.5 m Calculate the seat force using: Fseat Fw force factor mv 2 r Fseat ___________ Fw Part E—Camel Back Radius of the track, r = 16.0 m Calculate the seat force using: Fw Fseat force factor mv 2 r Fseat ___________ Fw Special thanks to Brent Ohl 84 The Hydra Questions: 1. 2. How do the force factors you experienced while riding compare to the force factors you calculated using Newton’s laws? Why are the values not exactly the same? The force factors experienced from the Camel Back and the Zero-g Roll are the same, but the orientation of the rider is very different. Why is this the case? Lab 2: Force Factor on the Spiral—The Banked Curve of This activity is designed for the Honors/Advanced Placement Physics student. The radius of curvature of the spiral is 16.1 m. Using the diagram below, draw the force diagram of the rider while on the spiral (Point F) described in the preliminary data section. Assume there are no forces applied to the rider that are parallel to the seat. Special thanks to Brent Ohl 85 The Hydra Front view of the rider Using the force diagram along with force factor and speed of the train you found in the preliminary data section, generate the equation for the banking angle of the spiral. Calculate the banking angle of the spiral: Banking Angle, θ = _______________________ According to your force diagram, generate another equation to calculate the banking angle of the spiral and solve for it. Banking Angle, θ = ____________________ Special thanks to Brent Ohl 86 The Hydra Analysis: According to the force diagram, we made the assumption that there are no parallel forces applied to the rider by the seat. This means that the banking angle of the track is perfect. When dealing with roller coasters, this typically does not happen. This way the train “searches” for equilibrium, and the train will wobble from side to side while traveling around the curve similar to that of a passenger train. So, engineers correct this problem by not making the banking angle perfect. Does your data verify this? Explain. Lab 3: Work/Energy considerations using Equipment needed: Stopwatch Inclinometer Power of the chain lift: Hold the inclinometer parallel to the lift hill and record the angle of the lift hill below: Angle of incline: _________________ Calculate the force of the chain lift, F|| to get the 12,560 kg train to the top of the hill assuming the train moves up the hill at constant speed. Special thanks to Brent Ohl 87 The Hydra F|| mg θ θ mg sinθ F|| = mg sinθ = ___________________ Measure the total time the front car of the train takes to make it up the lift hill. Time, t = ______________________ Calculate the average speed v of the lift hill chain knowing the hill is 69.5 m long. d _________________ t Calculate the amount of work done by the chain to lift the train up the hill. v= W = F|| d = ______________________ Calculate the power output of the chain lift motor in watts and horsepower knowing there is 746W/hp. P= W _______________Watts = ___________________hp t Compare the power found above to the power calculated by using the following equivalent equation: P = F|| v = ______________________ Total Energy of the train at various locations along the track (NOTE: All track height measurements will be in reference to the bottom of the first hill) Keeping in mind that gravitational potential energy is represented by the following equation: PE = mgh Special thanks to Brent Ohl 88 The Hydra and kinetic energy is given by the equation: 1 2 mv 2 Where mass m is the mass of the train. KE You can now find the total energy at any given point on the track by TE = PE + KE Calculate the energy at the top of the lift hill: The lift hill is 32.1m. PE = _________________________ KE = __________________________ TEinitial = ______________________________ Calculate the energy at Point A: The height at A is 0 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ TEA = _________________________ TEinitial - TEA = ______________________ Calculate the energy at Point B: The height at B is 20.2 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ TEB = _________________________ TEinitial – TEB = ______________________ Calculate the energy at Point C: The height at C is 5.5 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ TEC = _________________________ TEinitial – TEC = ______________________ Calculate the energy at Point D: The height at D is 8.5 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ Special thanks to Brent Ohl 89 The Hydra TED = _________________________ TEinitial – TED = ______________________ Calculate the energy at Point E: The height at E is 10.1 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ TEE = _________________________ TEinitial – TEE = ______________________ Calculate the energy at Point F: The height at F is 6.1 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ TEF = _________________________ TEinitial – TEF = ______________________ Calculate the energy at Point G: The height at G is 7.3 m above the bottom of the lift hill. PE = _________________________ KE = __________________________ TEG = _________________________ TEinitial – TEG = ____________________ The work energy theorem Wbrakes = KE can be used to calculate the average force needed to stop the train when it reaches the braking section of the ride. The kinetic energy of the train just before the brakes are applied is the KE at Point G. KEG = ________________________ The kinetic energy of the train when the train stops is KEstop = _______________________ Calculate the amount of work needed to stop the train: Wbrakes = _____________________ Special thanks to Brent Ohl 90 The Hydra Calculate the average force applied to the train by the brakes knowing W = Fd and the distance the brakes are applied to the train is 6.2 m. Fbrakes = _______________________ Questions: 1. Is the total energy the same at every point on the track you measured? Should it be the same—Explain. 2. How much energy was lost on the ride? What is the cause of this loss of energy? 3. It was mentioned previously that the brakes apply an average force to the train. Explain why it is an average force and not an instantaneous force. 4. You calculated the total amount of energy of the train at the top of the lift hill. Where did that energy come from? Critical Thinking Problem: Try to calculate the average frictional force applied to the train starting at the top of the lift hill to Point G given the length of the track being approximately 810 m. Special thanks to Brent Ohl 91 The Hydra Lab 4: The JoJo Roll of. . . This is a short conceptual activity using the JoJo Roll of the Hydra. The JoJo roll is the first element when leaving the station. This element is quite unique to roller coasters. Part 1 Go to the Cobra Roll side of the ride facing the "happy face" to get a front view of the train passing through the JoJo Roll. Question: 1. Estimate your force factor just before entering the roll. Explain your reasoning. Special thanks to Brent Ohl 92 The Hydra 2. Estimate your force factor when you are upside down in the roll. Explain your reasoning. Special thanks to Brent Ohl 93 The Hydra Part 2: While riding through the JoJo Roll, hold one vertical accelerometer upside down, hold one horizontally left, hold one horizontally right, and hold one right side up. Questions: 1. What was the difference in force factor during the JoJo Roll? Explain your reasoning. 2. Compare the JoJo roll to other roller coasters that go upside down with regards to the weightless feeling and the force factor. 3. Since you are going upside down, can you do something to the ride to create a weightless feeling on the JoJo roll? Why/Why not. 4. The JoJo is considered a heartline roll. By watching the train pass through the roll, explain the meaning of heartline roll. Special thanks to Brent Ohl 94 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Part I: - Directions: Using Graph 1 and 2 below answer the questions found below the graphs. Graph 1: Dominator: Shot Downward Questions: 1. A. If you and a friend were watching and waiting in line to ride the Dominator, at what point of the drop would you tell him/her on the ride that they will experience a feeling of weightlessness? B. At what time interval does this occur at according to the graph? ____________________________________ 2. If zero altitude is your starting position on the Dominator, according to the graph how high does the Dominator climb before dropping you? __________________________________ 3. The same friend you advised in question #1 is afraid that the Dominator is going to drop straight to the ground. According to graph 1, how much distance is between the ground and the lowest point on the first drop? __________________________________ 95 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. 4. From the graph of The Dominator, notice that the ride lasts longer than 120 seconds. From your interpretation of the first 120 seconds of the ride, and after looking at the ride in the park, draw what you think the altitude and the xacceleration due to gravity would look like if the graph actually took into account the ENTIRE ride. Hint: Take a stop watch and see how much longer the ride actually goes and how many more up and down motions the ride will experience after the 120 seconds represented on the graph. Graph 2: Revolution 5. According to graph #2 (Revolution), how many revolutions actually occurred in the 120 seconds? ______________________________________________ 6. What is the correlation between the altitude and the G forces acting on the xaxis according to the data obtained from Revolution? _______________________________________________ 7. If you think of Revolution as a pendulum, at what point would you experience the greatest G forces? The highest peak or at the lowest point in the ride? ___________________________________ 96 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Explain how you determined this by using specific references to the graph. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ________________________________________________ 8. After viewing the ride “Revolution” compare the graph to the actions of the actual ride. Notice again that the entire ride is not present on the graph. If “Revolution” runs an identical path each time it operates, do you think the gforce shown on the graph has reached its highest peak? If not, how many more peaks would there be before it reaches its highest peak? ______________________________________________________________________________ ______________________________________________________________________________ ____________________________________________________________ Part II Directions: For the following Graphs A – J, match the graph with the ride in Dorney Park. Answer the questions after all the graphs. Graph A 97 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Graph B Graph C 98 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Graph D Graph E 99 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Graph F Graph G 100 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Graph H Graph I 101 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. Graph J Dorney Park Ride Matching Graph Letter Apollo 2000 Wave Swinger Thunderhawk Steel Force The Hydra: Revenge Talon Enterprise Music Express Dominator: Being Shot Up Sea Dragon ____________________________ ____________________________ ____________________________ ____________________________ ____________________________ ____________________________ ____________________________ ____________________________ ____________________________ ____________________________ Analysis 1. Write a brief statement that describes your reasoning for selecting the graph you did for Thunderhawk? Were there specific details on the graph that made you 100% sure that this graph was the graph for Thunderhawk? ____________________________________________________________________________________ ____________________________________________________________________________________ ________________________________________________ 102 Interpreting Graphs Note: Alt on the graph represents altitude and the z axis is the vertical or “up and down” axis of acceleration. 2. Which graphs to rides interpretations were the hardest to make? For what reasons were they the most difficult? ____________________________________________________________________________________ ____________________________________________________________________________________ ________________________________________________ 3. Which rides had the most similar graphs? Why do you think they have similar graphs? ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________ 4. On graph I, what trend do you notice about the G forces when the altitude is at its peak versus when the altitude has reached its low point? ____________________________________________________________________________________ ____________________________________________________________________________________ ____________________________________________________________________________________ _________________________________ 103 Possessed Preliminary Data: Your weight in pounds = ________________lb X 4.45 N/lb = ___________________N weight in Newtons ___________________kg 9.8 m sec 2 Mass of loaded train = 13,065 kg Length of the train = 15.75 m Your mass, m in kilograms = Measure the time it takes for the train to do its first launch (L1) from rest. Suggestion: Use the center of the train as your reference point. This is the time it takes the train to be accelerated by the LIMs (linear induction motors). You may want to stand back from the ride to get a better overall view of the station to take your measurements. tL1 = ____________________sec Measure the time it takes for the train to pass through the second boost which is the same time it takes to pass through the station after the first launch. here (x = 60 m) here tL2 = ____________________sec 104 Possessed Measure the time it takes for the train to pass through the first braking pass in the station. tB1 = ____________________sec Measure the time it takes for the train to pass through the second braking pass in the station. tB2 = ____________________sec Section 1: Linear Acceleration 1. Since the distance that the train is traveling during its first launch is 48 m and the train starts from rest, use the linear equation given to calculate the average constant acceleration of the first launch. Remember that you measured the time. 1 x vi t at 2 2 Acceleration of the launch, a = ____________________m/sec2. 2. Assuming no friction, calculate the average net force you experience. Favg = ma _______________________N 3. Calculate the average amount of work the LIM’s do on you during the launch knowing the distance traveled during the launch given in #1. W = Fdcos______________________J 105 Possessed 4. Calculate your force factor you experience while launching. ff = 5. Favg weight ______________________ Using a linear equation given, calculate the speed of the train upon reaching the vertical section of the track. v 2f vi2 2ax OR v f vi at vL1 = _________________________m/sec 6. For the students with the CENCO lateral accelerometers, get in line and ride POSSESSED. While in line, secure and familiarize yourself with the lateral accelerometer. When on the ride, have your lateral (horizontal) accelerometer ready for launch. Position the accelerometer as instructed by your teacher. Measure the average lateral force factor of the first launch by recording the average location of the BB’s in the tube. ff = ____________________ Alt. 6. For students with the PASCO or handmade lateral accelerometers, get in line and ride POSSESSED. While in line, secure and familiarize yourself with your version of the inclinometer (used as a lateral accelerometer). When on the ride, have your inclinometer ready for launch. Position the accelerometer as instructed by your teacher. Measure the average angle of the first launch. Calculate the accelerating force, FN on you by using the following: θ θ Fnety = may FN cosθ – w = may FN cosθ – w = 0 FN cosθ = w w 106 Possessed Calculate your force factor while launching. ff = FN _____________________ weight Compare your results with that calculated in #4 elaborating on reasons for errors. Section 2: Work/Energy considerations for the Launch and Boost 7. Using the conservation of energy and the speed of the train after the initial launch found in procedure 5, calculate the height of the train on the spiral section of the track. KEL1 = PEvert. 1 2 mv L1 mghvert 2 h = _________________________m 8. Using the work/energy theorem and the values found in #6, calculate the average amount of work the LIM’s do on you during the first launch using the WLIM = KE 1 1 WLIM mv L21 mv i2 2 2 W = _______________________J 107 Possessed 9. Now calculate the average amount of work the LIM’s do on the train during the first launch. WLIM = KE 1 1 WLIM mv L21 mv i2 2 2 W = _______________________J 10. Calculate the average power delivered by the LIMs to the train during the first launch. P= W _______________________W t L1 OPTIONAL: 11. Calculate the average current supplied by the LIMs during the launch. The voltage provided to each LIM during launch is 240 V and assuming there are 4 LIMs operating at any given time. (NOTE: This is an over simplified version as to what actually is occurring electrically during the launch.) P = current x voltage Current = _______________________amperes 12. Assuming no friction, the speed of the train at the end of the first launch must the same as the speed of the train at the beginning of the second boost because of the conservation of energy. Knowing the speed of the train at the end of the first launch given in procedure 5, the distance (xL2 = 60.0 m) of the train during the boost (see picture on page 1), and the time it takes for the second boost from the initial data, calculate the acceleration and speed of the train after the second boost using the given linear equation. 1 x L 2 v L1t at 2 AND v L 2 v L1 at 2 aL2 = _________________________m/sec2 and vL2 = _____________________m/sec 108 Possessed 13. Using the work energy theorem, calculate the average amount of work on the train to accelerate it through the second boost. WLIM = KE 1 1 WLIM mv L2 2 mv L21 2 2 W = _______________________J 14. Calculate the average net force on the train for the second boost using the equation. Fnet ma OR W F (x L 2 ) cos F = _______________________N 15. Compare the force of the second boost to that of the first launch. Are they same or different and why? 16. Using the conservation of energy and the speed of the train after the second boost, calculate the height of the train on the straight vertical section of the track. KEL1 = PEvert. 1 2 mv L 2 mghvert 2 h = _________________________m 109 Possessed Section 3: The Vertical Braking The train gets stopped for 1 second by standard mechanical clamp brakes when it reaches its highest point of approximately 37 m as measured from the center of the train on the straight vertical section of the track. 17. How energy does the train have while held stationary by these brakes at this height? PE train mgh W = ________________________J 18. How much power is delivered by the brakes at this point? PE train P t P = _________________________W 19. How much force must the brakes be applying to the train to keep it held in this vertical position? Force = ________________________N Section 4: Stopping the train at the end of the ride 20. Using the conservation of energy, calculate the speed of the train upon entering the station after the vertical brake. PE vert.brake KE station mgh vert.brake 1 2 mv station 2 vstation = ___________________________m/sec 110 Possessed 21. You know the distance and the time, tB1, for the train to pass through the braking pass while passing through the station. You also know how fast the train is moving upon entering the station along with the initial speed upon entering the station from procedure 19. Calculate the acceleration and speed of the train at the end of the first braking pass through the station by using the equations. 1 xstation vstationt at 2 AND vB1 vstation at 2 aB1 = _____________________m/sec2 and vB1 = _______________________m/sec 22. Calculate the amount of work required to slow the train in the first braking pass. WLIM WLIM = KE 1 1 2 mv B21 mv station 2 2 WB1 = ________________________J 23. Calculate the average force the LIMs apply to the train to slow it down during the first braking pass. Fnet ma OR W F (x) cos Favg = __________________________N 111 Possessed 24. Since the train stops during it second braking pass through the station, calculate the amount of work required to slow the train in the second braking pass. WLIM = KE 1 WLIM 0 mv B21 2 WB2 = ________________________J 25. Calculate the average force the LIMs apply to the train to slow it down during the second braking pass. W F x cos Favg = __________________________N 26. Compare the work done or force applied by the LIMs to launch the train at the beginning of the ride to that in braking the train. Be sure to explain your reasoning. Section 4: Vertical Sections of the Ride . In this section of the lab, you will be looking at the vertical sections of the track to see if the spiral vertical section of the track affects freefall. Secure the vertical accelerometer as shown in the diagram below. This orientation is for when you are on the spiral vertical section of the track. Reverse the accelerometer for the straight vertical section of the track. You will need to orient the accelerometer so that the weight hangs suspended by the spring when the train is on these sections of the track. 112 Possessed Spiral side straight side When the train is at the highest on these sections of the track, record the force factor. Be sure you are not recording the force factor while the train is being stopped on the straight vertical section of the track. If you do not have a vertical accelerometer, you can use a stopwatch to measure the time it takes for the train to freefall on these sections. Vertical Spiral section of track ffspiral or time = ________________ Vertical straight section of track ffstraight or time =________________ Explain any differences in the force factor on these two sections of track. Should there be a difference? Why or why not? Section 5: The Upward Curved Sections and Centripetal Force In this section of the lab you will calculate the radius of each vertical curve by using the force factors and speeds that you measure. This section can be done as a stand-alone activity or can utilize the 113 Possessed data from the previous sections. You can use the speeds from the previous sections. It would be helpful if you have several students taking measurements at the same time. Measure the time the train takes to pass the lone vertical support post outside of the station on each side of the station shown in the picture. If time permits, measure the speed at these points for one, two, or three passes. The more data you have, the more accurate your calculations. Enter the data into the table below. By knowing the length of the train given in the preliminary data at the beginning of the lab, calculate the speed of the train just as it enters the vertical curves. Enter the data in the table below. Secure the vertical accelerometer and measure the force factor you experience while passing through the curves for each pass you predetermined. Be sure you are measuring the force factor that corresponds to the time you measured. Also, be sure that you are holding the vertical accelerometer parallel to your body as shown in the picture on the next page. Spiral side Pass time (sec) speed (m/sec) force factor Vertical Side radius (m) time (sec) speed (m/sec) force factor radius (m) 1 2 3 114 Possessed According to the free-body diagram and the two expressions given, Fseat ac ff Fseat w Fseat w mv 2 r w 1. Derive an expression for calculating the radius of curvature of the track. 2. Calculate the radius of curvature for each curve and enter the data into the table on the previous page. QUESTIONS: 1. Compare the radius on the straight side and the spiral side of the ride. 2. As the speed increased, explain what happened to the force factor. 3. As the speed increased, explain what happened with the radius? 4. If the radius was smaller, what would happen to the force factor? Speed? Support your answer. 115 Demon Drop Introduction: This experiment consists of three parts. Part one will investigate the free-fall portion of the ride. Part two will analyze the ride from a work, power, and energy point of view. Part three will demonstrate the effects of friction during the braking period of the ride. Equipment Needed: Stopwatch Vertical Accelerometer Refer to the following diagram while completing this activity: B C 13.56 m D 34.43 m 19.25 m E F A Variables 𝑣 - velocity 𝑣𝑓 – final velocity 𝑣𝑖 – initial velocity 𝑎 – acceleration 𝑡 – time ∆𝑦 –vertical displacement 𝑊 – work 𝐹 – force 𝑑 – displacement 𝜃 – angle between the force and displacement vectors 𝑇𝐸 – total energy 𝐾𝐸 – kinetic energy 𝑃𝐸𝑔 –potential energy 𝑃 – power 𝑚 – mass of the car and its passengers Σ𝐹 – the net force acting on the car 116 Demon Drop PART 1 (Free-fall) Useful Formulae: 𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡 % 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 100 ( 𝑣𝑓 2 = 𝑣𝑖 2 + 2𝑎∆𝑦 ∆𝑦 = 2 (𝑣𝑖 + 𝑣𝑓 )𝑡 |𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑−𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑| 𝑎𝑐𝑐𝑒𝑝𝑡𝑒𝑑 ) 1 1 ∆𝑦 = 𝑣𝑖 𝑡 + 2 𝑎𝑡 2 1. Using the stopwatch, measure the time it takes for the car to drop from the top of the ride to the point immediately before the track curves (from point C to D in the diagram). Repeat the measurement several times and find the average time. Trial Time, t (s) 1 2 3 Average 2. a. Using the average time measured in step 1 and the appropriate height from the diagram, calculate the acceleration, 𝑎, of the car as it travels from C to D. b. Compare this acceleration to the accepted value of g, 9.8 m/sec2, with a percent difference. 3. Now use -9.8 m/sec2 for the free-fall acceleration, while also using the average time value, 𝑡, to calculate the velocity, 𝑣𝑓 , of the car at point D. 117 Demon Drop 4. Now use the vertical accelerometer on the ride to measure the force factor (g-force) experienced by the rider while in free-fall. Force factor = Conclusion Questions 1. Was your calculated value of acceleration due to gravity in step 2 larger or smaller than the accepted value of g? Why do you think this is the case? 2. In step 4, was the measured value of force factor what you expected? Why or why not? PART 2 (Work, Power, and Energy) Useful Formulae: 𝑊 𝑊 = 𝐹𝑑 cos 𝜃 𝑃= 𝑇𝐸 = 𝐾𝐸 + 𝑃𝐸𝑔 𝐾𝐸 = 2 𝑚𝑣 2 𝑡 1 𝑃𝐸𝑔 = 𝑚𝑔ℎ 1. The Demon Drop car has a mass of 858.2 kg and rises from point A to its maximum height at point B. If the car seats four people that each have a mass of 60 kg, what is the gravitational potential energy of the car and its passengers at the top of the ride? 118 Demon Drop 2. Using the conservation of energy and assuming friction is negligible, use the energy calculated in step 1 as total energy (TE) to calculate the velocity of the car and its passengers at point D. 3. Sketch a graph showing the relationship between kinetic energy, potential energy, and total energy versus time as the car travels from point C to E. Be sure to label the axes, with units included. 4. a. Using the height of the tower, calculate the work done by the motor while raising the car from point A to B. Assume the car is raised at a constant velocity. 119 Demon Drop b. Now calculate the work done by gravity while raising the car from point A to B. 5. Using the stopwatch, measure the time it takes for the car to elevate from point A to B. Complete three trials and find the average time. Trial Time (s) 1 2 3 Average 6. Use the average time from step 5 to calculate the average power in kilowatts needed to elevate the car. 7. If the ride lifts the car 100 times per hour, how much would it cost to operate the ride for one hour given that the price of electricity is $0.12 per kWh? (Use the power from step 6 and the average time from step 5 as the power and time it takes to lift the car once) 120 Demon Drop Conclusion Questions 1. How much faster would the car be going if each passenger had a mass of 80 kg instead of 60 kg for step 2? 2. Observing the graph from step 3, how does total energy change over this interval and why? 3. If it took only half the time to lift the car from point A to B, by what factor would the power change, assuming the mass of the car and its passengers remains unchanged? 4. Compare the velocity found in step 2 with the velocity found in the third step of Part 1. Which value do you think is closer to the actual value and for what reason? PART 3 (Braking) Useful Formulae: Σ𝐹 = 𝑚𝑎 Equation 1: 𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡 1 Equation 2: ∆𝑥 = 𝑣𝑖 𝑡 + 2 𝑎𝑡 2 1. Draw a free body diagram of the car at point E. Be sure to include the force of gravity, the reaction force to gravity, and the force causing acceleration. Also draw the velocity and acceleration vectors separate from the free body diagram. 121 Demon Drop 2. Using the stopwatch, measure the total braking time for the car, starting when the car begins to brake, at point E, and ending when the car comes to a complete stop, at point F. Complete three trials and find the average time. Trial Time (s) 1 2 3 Average 3. Use the average time measured in step 2 and the distance between points E and F to calculate the average acceleration over this displacement. Assume there is uniform braking and that 𝑣𝑓 = 0. (Hint: Substitute Equation 1 into Equation 2; the velocity from step 2 cannot be used here, because the velocity at point D is not equal to the velocity at point E) 4. a. Use the answer from step 3 and Newton’s Second Law to calculate the force of friction necessary to bring the car to a complete stop. (Use the mass from Part 2) b. Calculate the work done by friction during braking (between E and F). 122 Demon Drop Conclusion Questions 1. Explain why the velocity and the acceleration vectors are in the directions that they are in the free body diagram. 2. What was the work required to lift the car from point A to B? What was the work needed to stop the car from point E to F? Should these values be the same? Why or why not? 3. Honors/AP Question: What is the difference between a conservative force and a nonconservative force? Which type of force is friction? I would like to acknowledge the assistance of two Kutztown University physics majors, Nate Benjamin and Kevin Ruppert. Their assistance in the development of the physics activities, The Demon Drop and Meteor proved to be very valuable. 123 Meteor Introduction: This experiment consists of three parts. Part one will investigate the circular motion of the ride as it pertains to centripetal force and angular acceleration. Part two will apply the concepts of oscillatory motion and torque to the path in which the ride travels. Part three deals with the force factor that you, the rider, experience at the top and bottom of the ride. Equipment Needed: Stopwatch Vertical Accelerometer Variables 𝑣 – tangential velocity 𝑟 – radius from axis of rotation 𝜔 – angular velocity/angular frequency 𝜔𝑖 –initial angular velocity 𝜔𝑓 – final angular velocity 𝑎 – tangential acceleration 𝛼 – angular acceleration 𝑎𝑐 – centripetal acceleration ∆𝜃 –angular displacement 𝑡 – time 𝑇 – time period 𝑓 – frequency 𝜏 – torque F – force 𝜙 – the angle between the force and radius ectors 𝑓𝑓 – force factor/g-force 𝐹𝑁 – normal force on you from the seat 𝑚 – your mass 𝑔 – acceleration due to gravity Σ𝐹 – the net force acting on an object 𝑤 – your weight PART 1 (Acceleration) Useful Formulae: 𝜋 1 𝜃𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 𝜃𝑑𝑒𝑔𝑟𝑒𝑒𝑠 (180°) ∆𝜃 = 2 (𝜔𝑖 + 𝜔𝑓 )𝑡 1 𝑓𝑡 = 0.3048 𝑚 𝜔𝑓 = 𝜔𝑖 + 𝛼𝑡 𝑣 = 𝑟𝜔 𝜔𝑓 2 = 𝜔𝑖 2 + 2𝛼Δ𝜃 𝑎 = 𝑟𝛼 Δ𝜃 = 𝜔𝑖 𝑡 + 2 𝛼𝑡 2 𝑎𝑐 = 𝑣2 𝑟 1 = 𝑟𝜔2 124 Meteor Use the following diagram to complete the activities in Part 1: C B A 𝑣 = 0 𝑚/𝑠 𝑣 = 0 𝑚/𝑠 𝑡𝑖 = 0 𝑠 𝑣 = 0 𝑚/𝑠 𝑡𝑖 = 0 𝑠 100° 135° 45° 𝑡𝑖 = 0 𝑠 𝑡𝑓 𝑡𝑓 𝑡𝑓 1. Meteor changes the direction in which it rotates on multiple, notable occasions – twice when the ride first begins in order to get up to speed, as seen in figures A and B, and once midway through the ride, displayed in figure C. Using the chart below, complete parts a. through f. Time, t (s) Initial direction change (Figure A) Second direction change (Figure B) Midway direction change (Figure C) 𝚫θ Δθ (rad) α (rad/s2) ωf (rad/s) v (m/s) ac (m/s2) 45° 100° 135° a. At each of these points, measure the time, 𝑡, it takes for either arm to go from its maximum height (when it is at rest) to the bottom (when it passes the vertical). Record the time in the chart above. b. The angular displacement, ∆𝜃, for each of these intervals is given in the third column of the above chart. Convert these angles given in degrees to angles in decimal radians and record them in the chart. Show a single sample calculation below (not for each angle). 125 Meteor c. Use the radian angles that you just calculated in 1b and the measured times in 1a to calculate the angular accelerations, 𝛼, of either arm during these intervals and record these data in the table above. Show one sample calculation below. d. Knowing 𝜔𝑖 = 0, use the data from the chart to calculate the angular velocity, 𝜔𝑓 , of either car at the bottom of the swing and record the data in the chart. Show one sample calculation below. e. The approximate distance from the axis of rotation to the seats is 35 feet. Convert this value and use the answers from 1d to calculate the tangential velocities, 𝑣, and record them in the chart. Show a sample calculation below. f. Calculate the centripetal acceleration, 𝑎𝑐 , at each point using the tangential velocities, 𝑣, or the angular velocities, 𝜔𝑓 , that were previously found. Record these data in the chart and show a sample calculation below. 126 Meteor Conclusion Questions 1. Were the values for the angular accelerations, 𝛼, similar during the different intervals? What does this imply? 2. What is the conceptual difference between centripetal acceleration and angular acceleration? 3. In accordance with Newton’s Second Law, what two forces are needed to calculate the net force (centripetal acceleration)? 127 Meteor PART 2 (Oscillations) Useful Formulae: 1 𝑇=𝑓= 2𝜋 𝜏 = 𝑟𝐹 sin 𝜙 𝜔 𝜔 = 2𝜋𝑓 1. In this portion of the lab, you will compare the oscillatory motion of the forward-rotation half of the ride with the reverse-rotation half of the ride. For consistency, all of the data in this section must be collected during a single run. Once the ride gets up to full speed, utilize the lap feature on the stopwatch to measure the time it takes for one of the arms to complete three full rotations, recording each time period, 𝑇. Repeat this process once the ride changes directions midway through the run. Cycle Forward, T (s) Backward, T (s) 1 2 3 Average 2. Using the averages from step 1 calculate the angular frequencies, 𝜔, of the forward and backward cycles. ωforward = ωbackward = 3. Compare the average forward time period with the average backward time period using a percent difference (use the average of the forward and backward time periods for your denominator). % 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 100 ( |𝑇𝑓𝑜𝑟𝑤𝑎𝑟𝑑 −𝑇𝑏𝑎𝑐𝑘𝑤𝑎𝑟𝑑 | 𝑇𝑎𝑣𝑒𝑟𝑎𝑔𝑒 ) 128 Meteor 4. Given that the radius from the axis of rotation to the seats is 35 feet and the mass remains constant during a single ride, at what two points would the torque on the car caused by gravity be zero? At what two points would it be at its maximum and in which direction is the torque? Sketch a torque vs. angular position graph, labeling the angles relative to the vertical at the bottom of the ride. Be sure to label the axes, with units included. Torque, 𝜏 (m·N) Angular Position, 𝜃 (radians) 0 −𝜏𝑚𝑎𝑥 0 𝜏𝑚𝑎𝑥 0 2π 𝜏𝑚𝑎𝑥 −𝜏𝑚𝑎𝑥 Conclusion Questions 1. How did the angular frequencies of step 2 compare with the angular velocities from the chart in Part 1? Were they similar or dissimilar? Why do you think this is? 129 Meteor 2. Assuming angular velocity remains constant while the ride is at full speed, does the tangential velocity change? Explain your reasoning? 3. In step 3, was the time period the same regardless of the direction? What does this show? 4. Hypothesize about why the ride contains two cars traveling in opposite directions rather than a single car? 5. Examine the graph from step 4. What common function does this graph appear to mimic? Use the equation for torque to explain this relationship. 130 Meteor PART 3 (Force Factor/g-Force) Useful Formulae: 𝐹 𝑁 𝑓𝑓 = 𝑚𝑔 Σ𝐹 = 𝑚𝑎𝑐 𝑤 = 𝑚𝑔 1. While on the ride, use the vertical accelerometer to measure the force factor (g-force) at the very top of the ride once it reaches its maximum speed. Also measure the force factor at the lowest point of the ride. 𝑓𝑓𝑡𝑜𝑝 = 𝑓𝑓𝑏𝑜𝑡𝑡𝑜𝑚 = 2. Using the force factor measurements from step 1, calculate the normal force exerted on you by the seat at each location. Assume your weight is 600 N. 3. 𝐹𝑁𝑡𝑜𝑝 = 𝐹𝑁𝑏𝑜𝑡𝑡𝑜𝑚 = 4. Draw a free-body diagram of yourself when you are at the top of the ride (first diagram) and when you are at the bottom of the ride (second diagram). 131 Meteor 5. Use Newton’s second law of motion to calculate the centripetal acceleration that you experience while at the top and while at the bottom. 𝑎𝑐𝑡𝑜𝑝 = 𝑎𝑐𝑏𝑜𝑡𝑡𝑜𝑚 = Conclusion Questions 1. What are the units for force factor? 2. At what location was force factor greatest, the top or bottom of the ride? Why? 3. Compare the centripetal acceleration at the top of the ride with the centripetal acceleration at the bottom of the ride. Should they be similar? If so, explain why. If not, what accounts for the difference? 132 Meteor 4. If the car is traveling counterclockwise around the circle, in what direction is the acceleration vector pointing when the car is at the top? At the bottom? In what direction is the velocity vector pointing in each of these locations? I would like to acknowledge the assistance of two Kutztown University physics majors, Nate Benjamin and Kevin Ruppert. Their assistance in the development of the physics activities, The Demon Drop and Meteor proved to be very valuable. 133