Thrill U. · PDF fileThrill U. THE PHYSICS AND MATHEMATICS OF AMUSEMENT PARK RIDES . Physics . ... the dynamics of the rides as a review, incorporate them within a laboratory practical,

Thrill U. THE PHYSICS AND MATHEMATICS OF AMUSEMENT PARK RIDES

Physics

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.

© Copyrighted by Dr. Joseph S. Elias. This material is based upon work supported by the National Science Foundation under Grant No. 9986753.

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 1-123

Apollo 2000 Page 1 The Antique Carrousel Page 8 The Ferris Wheel Page 13 The Enterprise Page 19 Revolution Page 23 The Dominator Page 27 The Sea Dragon Page 32 White Water Landing Page 37 The Scrambler Page 42 The Wave Swinger Page 47 Steel Force Page 52 Energy Curves for Steel Force Page 61 Centripetal Force and Steel Force Page 65 The Talon Page 67 Thunderhawk Page 75 The Hydra Page 79 Interpreting Graphs Page 95 Possessed Page 104 Demon Drop Page 116 Meteor Page 124

Thrill U.

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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 Allentown School District Mr. Patrick Callahan Delaware Regional School District Mr. Bernie Bonuccelli Dorney Park of Allentown Mr. Joseph Greene Dorney Park of Allentown Mr. Keith Koepke Dorney Park of Allentown Mr. Edward Anthony East Penn School District Mr. Brent Ohl East Penn School District Ms. Carole Wilson East Penn School District Dr. David Drummer Kutztown Area School District Mr. Richard Button Kutztown University of Pennsylvania Dr. Kathleen Dolgos Kutztown University of Pennsylvania Dr. Joseph Elias Kutztown University of Pennsylvania Dr. Deborah Frantz Kutztown University of Pennsylvania Dr. Neal Shea Kutztown University of Pennsylvania Ms. Brenda Snyder Kutztown University of Pennsylvania Mr. Glenn Frey Northwestern Lehigh School District Mr. Jeffrey Wetherhold Parkland School District Mr. Jeffrey Bartman Parkland School District Ms. Brandi Murphy Parkland School District Mr. Gerry Farnsworth Parkland School District Mr. Robert Guigley Reading Area School District Ms. Maggie Woodward Upper Perkiomen School District

…and to the many graduate students of the Kutztown University of Pennsylvania who contributed to the development of this manual.

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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] Dr. Joseph S. Elias Kutztown University [email protected] of Pennsylvania

mailto:[email protected]

mailto:[email protected]

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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.. or

visit our website: www.dorneypark.com

Thank you.

http://www.dorneypark.com/

Apollo 2000

1

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 = =5t _________________sec

3. The angular velocity, ω, is the angle that is swept out over a period of time of a

rotating object. Calculate the angular velocity. Remember one rotation is 2π radians angular distance.

ω = __________________ rad/sec

Apollo 2000

2

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.

θmax 4.73 meters θmin 8.4 meters across center Inclinometer Inclinometer θmin = ___________________degrees θmax = ___________________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=ω r.

linear velocity at minimum radius = vmin = _________________ m/sec linear velocity at maximum radius = vmax = _______________ m/sec

Apollo 2000

3

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)

θmin

max21θ

max43θ

θmax

Apollo 2000

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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)

Apollo 2000

5

• 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!

θ FN

θ Fc Equation to use FW = mg

c

c

cN

N

N

ag

mamgmaF

andmgF

mgF

=

=

=

=

=

θθ

θθ

θ

θ

tancos

sinsin

,cos

cos

centripetal acceleration at θmin = ac(min) = ___________________ m/sec2 centripetal acceleration at θmax = ac(max) = ___________________ m/sec2

Apollo 2000

6

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

Answers from Procedure #11

GROUPS


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.

Apollo 2000

7

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?

The Antique Carrousel

8

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) _________


9

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 π 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*π∗5.8 meters / T=_______________________

(5.8 meters is the radius of the inner row of horses) 2. V2= 2*π∗6.6 meters / T=_______________________

(this row radius is 6.6 meters) 3. V3= 2*π∗7.3 meters / T=_______________________ 4. V4= 2*π∗8.1 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.


10

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 θ. 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


11

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:

ω=2∗π (radians)/T (seconds).

• What is the angular velocity for the carrousel?

ω=_______________________

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=ω*R.

For example V1=ω∗5.8 meters.

• How does the angular acceleration compare to angular velocity?

If we start with ac=V2/R, and substitute V=ω∗R, we prove ac =ω2*R.


12

As you see, there is a linear relationship between acceleration, a, and radius, R. Graph this relationship below using the value for ω 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?

The Ferris Wheel

13

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 (2πR). Using the radius indicated, calculate the velocity from:

v = 2 π 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

The Ferris Wheel

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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) = _______ Your Weight (N) = m * 9.8 m/s/s = _______ 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.

velocity (m/s) Radius of Wheel (m)

Acceleration (m/s/s)

12.3 Centripetal

Force (N)

The Ferris Wheel

15

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. 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

Weight Fg

Seat Force Fs

Seat Force Fs

Seat Force Fs

Weight Fg

Weight Fg

Weight Fg

Seat Force Fs

SeatBack Force FB

Seat Back Force FB

A

B

C

D

The Ferris Wheel

16

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

A C

Position Fw (N)

Your Weight Vertical F C (N)

Fs (N)

Seat Force B 0 D 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:

The Ferris Wheel

17

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 Trial 1 g force Trial 2 g force Trial 3 g force Average “g”

force

Location B Trial 1 g force Trial 2 g force Trial 3 g force Average “g”

force

Location C Trial 1 g force Trial 2 g force Trial 3 g force Average “g”

force

Location D Trial 1 g force Trial 2 g force Trial 3 g force Average “g”

force

The Ferris Wheel

18

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.

The Enterprise

19

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.

The Enterprise

20

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

____________

___________

____________

The Enterprise

21

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. Calculate the centripetal acceleration using ac = 4π

2 r / T

2

ac = 4π

2 r / T

2

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: __________________

The Enterprise

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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: ______________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________

Revolution

23

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.

Revolution

24

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.

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

A B

Revolution

25

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

Revolution

26

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 = ________________________________

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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. 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) Vup=-Vdown ∆V = Vdown - Vup = 2 Vup = g Tfreefall

Sketch Dominator and mark in relative positions and time data.

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)

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?

29

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)

30

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.

31

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.

The Sea Dragon

32

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, θ, that the ride makes with the vertical.

Maximum angle, θ _________________

The Sea Dragon

33

(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 - cosθ) 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.

The Sea Dragon

34

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_________________

The Sea Dragon

35

(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_________________

The Sea Dragon

36

(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 =2π(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?

White Water Landing

37

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 mass (x) gravitational field (x) height

(Top of the incline) • Kinetic Energy KE = 1/2 mv2 1/2 mass (x) velocity squared (Bottom of the incline)

• gravitational field g = 9.8 m/s/s at Dorney Park • height of incline 25 meters • Potential Energy Joules (J) • Length of Boat 5.2 m

Part (a) Velocity immediately before splashdown

White Water Landing

38

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 25 meters gravitational field 9.8 m/s/s Potential Energy at top of hill (J) =_______________ Kinetic Energy at bottom of hill (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)

White Water Landing

39

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 above-mentioned 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

White Water Landing

40

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

White Water Landing

41

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.

The Scrambler

Laboratory courtesy of Andrew Snyder, Materials Engineering, Rensselaer Polytechnic Institute Page 42

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.

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): ________________________________

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 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.

The Scrambler


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: __________________ Radius of the sweep arm at the point of minimum speed (distance of the car from the point of rotation of the sweep arm): _____________________

The Scrambler


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 Pivot arm from pivot to car: 1.82 meters

The Wave Swinger

47

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? ________________________

The Wave Swinger

48

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?

The Wave Swinger

49

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?

____________________________________________________

The Wave Swinger

50

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 θ) where θ 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.

The Wave Swinger

51

Analysis:

1. Calculate the predicted acceleration using g(tan θ), 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 = 4π2r/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 θ).

52

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.

Steel Force

53

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.

Steel Force

54

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

Line 2

VC=19.6m/____________ seconds

=

______________ units

Line 3

VE=19.6m/____________ seconds

=

______________ units

Line 4

VB=34 m/____________ seconds

=

______________ units

Line 5

VD=π*62 m/_________ seconds

=

______________ units

Line 6

Ahill=∆V/t =(________-_________)/____________

=

______________ units

Line 7

Ahill=∆V/t =(________-_________)/____________

=

______________ units

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.

Steel Force

55

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 sinθ. 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 name of the other (12) E (total mechanical energy) = _____________+____________ = 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. (13) P at B = _____________ (14) P at C = _____________

Steel Force

56

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

Steel Force

57

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 Mg principles using the physics and trigonometry studied in class.

When you are traveling in the train, your body is Fc 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.)

Steel Force

58

Points B, C and E are other interesting positions. In these areas, you have a combination of 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. 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:

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

gravity seat force

ΣF = m a

59

Data Taking Sheet for Side 1, Kinematics

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. t =_______________

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 =_________________

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. t =________________

Data Point C, The velocity at the top of the second hill is again found using train length and Time, t, =_________________

60

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. a=________________

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=________________

Energy Curves for Steel Force

Special thanks to Jeff Wetherhold 61

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.



Data:

Diagram 1 Length of train = 19.6 meters Rider’s weight = _____ pounds Rider’s mass = ______ kilograms

20 m

A, 0 m B

D E

F G H

I

J

K

L

height of hill = 46

C



Position mark

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) A 0 B C D E F G H I J K L



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?

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 “g-force” 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:

Diagram 1

20 m

TOP

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.

7. Knowing that acceleration of the rider at the top is given by the equation ac = rv 2

,

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 cF

= māc . Use this fact to determine the

centripetal force on the rider at the top. 9. Using a % difference, compare the centripetal forces you found in analysis # 5 and # 8.

67

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 = Weight in pounds/2.2 Magnitude of velocity = Distance traveled/time interval or v = d/t Centripetal acceleration = Velocity squared divided by radius or aC=v2/r Centripetal Force = Mass times centripetal acceleration or F = mac Force due to gravity (weight) =

Mass times gravitational acceleration or Fg=mg (g = 9.8 m/s/s at Dorney Park)

Force Factor = Seat Force/Force of Gravity or FF = Fs/Fg

68

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



At the peak of the Zero “g” Roll Length of Train = 12.2 m



In the middle of the horizontal Spiral Length of Train = 12.2 m



69

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 velocity (from part 1) = ______________ Centripetal Acceleration = ____________ Centripetal Force = _________________ Top of the vertical loop (when you are upside down): Radius of curve= 6.0 m velocity (from part 1) = _____________ Centripetal Acceleration = ____________ Centripetal Force = ________________ At the peak of the Zero “g” Roll Radius of curve= 18. 0 m velocity (from part 1) = _____________ Centripetal Acceleration = ____________ Centripetal Force = ________________ In the middle of the horizontal Spiral Radius of curve= 9.1 m velocity (from part 1) = _____________ Centripetal Acceleration = ____________ Centripetal Force = ________________

70

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 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):


Fg

Fs

Bottom of loop. Fc is up, so Fc=Fs-Fg

Seat Force Fs = Fc+Fg

Fs

Fg

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

71

At the peak of the Zero “g” Roll


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.

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):




In the middle of the horizontal Spiral


Force holding you up = your weight (Fg)

Centripetal Force (Fc)

Seat Force (Fs)

72

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 = ______________


Measured “g” Force = ______________

In the middle of the horizontal Spiral

Measured “g” Force = ______________

73

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.

Vertical Loop

Bottom

of First Hill

Horizontal Spiral

Zero “g” R

oll

Imm

elman

Inclined Spiral

Thunderhawk An Enlightening Lab

75

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.

Figure 1

Red dot

Top of hump

Top of hill

6 ft


76

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:

2. Top of the hump:

3. Fraction of Mechanical Energy Remaining

4. Percentage of Mechanical Energy Remaining

Total MechanicalEnergy= PE + KE = mgh +1

2mv

2

Total MechanicalEnergy= KE =12

mV2

=

12

mV2

mgh +12

mv2

=

12

V2

gh +12

v2(mass cancels)

=V

2

2 gh + v2(multiplyby two)

=V

2

2gh + v2

Χ 100


77

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.


78

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

The Hydra

Special thanks to Brent Ohl 79

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

The Hydra


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.

A

B

C D

E

F

G

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.

The Hydra


Knowing the that the length of the train is 12.298 meters long, ∆x, and txv

∆∆

= , calculate the speed of the

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 lbs

N = __________________N

Your mass in kg, m = =g

Fw ______________________ kg

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: r

mvFF wseat

2

=−

___________FF factor force

w

seat ==

Part B – top of zero-g roll Radius of the track, r = 16.0 m


mvFF seatw

2

=−


w

seat ==

Part C – After zero-g Roll Radius of the track, r = 20.8 m


mvFF wseat

2

=−

The Hydra



w

seat ==

Part D—Cobra Roll (a.k.a. Happy Face) Radius of the track, r = 15.5 m


mvFF wseat

2

=−


w

seat ==

Part E—Camel Back Radius of the track, r = 16.0 m


mvFF seatw

2

=−


w

seat ==

The Hydra


Questions: 1. 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?

2. 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?

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.

Lab 2: Force Factor on the Spiral—The Banked Curve of

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 = ____________________

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.

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.

Lab 3: Work/Energy considerations using

The Hydra


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.

v = _________________=td

Calculate the amount of work done by the chain to lift the train up the hill. W = F|| d = ______________________ Calculate the power output of the chain lift motor in watts and horsepower knowing there is 746W/hp.

P = =t

W _______________Watts = ___________________hp

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

mg

mg sinθ

F||

θ

θ

The Hydra


and kinetic energy is given by the equation: 2

21 mvKE =

Where mass m is the mass of the train. 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 = __________________________

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 = _____________________

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.

The Hydra


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.

Lab 4: The JoJo Roll of. . .

The Hydra


2. Estimate your force factor when you are upside down in the roll. Explain your reasoning.

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.

Interpreting Graphs Note: Alt on the graph represents altitude and the

z axis is the vertical or “up and down” axis of acceleration.

95

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?

__________________________________



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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 x-acceleration 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 x-

axis 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? ___________________________________



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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 g-force 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



98

Graph B

Graph C



99

Graph D

Graph E



100

Graph F

Graph G



101

Graph H

Graph I



102

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? ________________________________________________________________________________________________________________________________________________________________________________________________________________________



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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? _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Preliminary Data: Your weight in pounds = ________________lb X 4.45 N/lb = ___________________N

Your mass, m in kilograms = =2sec8.9Newtonsin weight

m___________________kg

Mass of loaded train = 13,065 kg Length of the train = 15.75 m 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

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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.

2

21 attvx i +=∆

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 = Fdcosθ______________________J

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4. Calculate your force factor you experience while launching.

ff = =weight

Favg ______________________

5. Using a linear equation given, calculate the speed of the train upon reaching the vertical section

of the track. atvvORxavv ifif +=∆+= 222

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: Fnety = may FN cosθ – w = may FN cosθ – w = 0 FN cosθ = w

θ

θ

w

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Calculate your force factor while launching.

ff = =weight

FN _____________________

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.

vertL mghmv =212

1

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 22

1 21

21

iLLIM mvmvW −=

W = _______________________J

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9. Now calculate the average amount of work the LIM’s do on the train during the first launch. WLIM = ∆KE

221 2

121

iLLIM mvmvW −=

W = _______________________J 10. Calculate the average power delivered by the LIMs to the train during the first launch.

P = =1Lt

W _______________________W

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.

atvvANDattvx LLLL +=+=∆ 122

12 21

aL2 = _________________________m/sec2 and vL2 = _____________________m/sec

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13. Using the work energy theorem, calculate the average amount of work on the train to accelerate it through the second boost.

WLIM = ∆KE 21

22 2

121

LLLIM mvmvW −=

W = _______________________J 14. Calculate the average net force on the train for the second boost using the equation.

θcos)( 2Lnet xFWORmaF ∆==

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.

vertL mghmv =222

1

h = _________________________m

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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?

mghPEtrain = W = ________________________J 18. How much power is delivered by the brakes at this point?

tPE

P train=

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.

2.

.

21

stationbrakevert

stationbrakevert

mvmgh

KEPE

=

=

vstation = ___________________________m/sec

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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.

atvvANDattvx stationBstationstation +=+=∆ 12

21

aB1 = _____________________m/sec2 and vB1 = _______________________m/sec 22. Calculate the amount of work required to slow the train in the first braking pass.

WLIM = ∆KE 22

1 21

21

stationBLIM mvmvW −=

WB1 = ________________________J 23. Calculate the average force the LIMs apply to the train to slow it down during the first braking

pass. θcos)( xFWORmaFnet ∆==

Favg = __________________________N

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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 2

1210 BLIM mvW −=

WB2 = ________________________J 25. Calculate the average force the LIMs apply to the train to slow it down during the second braking

pass. ( ) θcosxFW ∆=

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.

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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 Vertical straight section of track ffspiral or time = ________________ 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

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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 Vertical Side

Pass time (sec)

speed (m/sec)

force factor

radius (m)

time (sec)

speed (m/sec)

force factor

radius (m)

1

2

3

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According to the free-body diagram and the two expressions given, Fseat ac

rmvwF

wF

ff

seat

seat

2

=−

=

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.

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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: Variables 𝑣𝑣 - velocity 𝑣𝑣𝑓𝑓 – final velocity 𝑣𝑣𝑖𝑖 – initial velocity 𝑎𝑎 – acceleration 𝑡𝑡 – time ∆𝑦𝑦 –vertical displacement 𝑊𝑊 – work 𝐹𝐹 – force

𝑑𝑑 – displacement 𝜃𝜃 – angle between the force and displacement vectors 𝑇𝑇𝑇𝑇 – total energy 𝐾𝐾𝑇𝑇 – kinetic energy 𝑃𝑃𝑇𝑇𝑔𝑔 –potential energy 𝑃𝑃 – power 𝑚𝑚 – mass of the car and its passengers Σ𝐹𝐹 – the net force acting on the car

A

B C

E

D

F

13.56 m

34.43 m 19.25 m

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PART 1 (Free-fall) Useful Formulae: 𝑣𝑣𝑓𝑓 = 𝑣𝑣𝑖𝑖 + 𝑎𝑎𝑡𝑡 % 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 100 �|𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐−𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐|

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐�

𝑣𝑣𝑓𝑓2 = 𝑣𝑣𝑖𝑖2 + 2𝑎𝑎∆𝑦𝑦 ∆𝑦𝑦 = 12�𝑣𝑣𝑖𝑖 + 𝑣𝑣𝑓𝑓�𝑡𝑡

∆𝑦𝑦 = 𝑣𝑣𝑖𝑖𝑡𝑡 + 12𝑎𝑎𝑡𝑡2

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, 𝑎𝑎, of the car as it travels from C to D. b. Compare this acceleration to the accepted value of g, 9.8 m/sec 2, with a percent difference.

3. Now use -9.8 m/sec 2 for the free-fall acceleration, while also using the average time value, 𝑡𝑡, to calculate the velocity, 𝑣𝑣𝑓𝑓, of the car at point D.

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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:

𝑊𝑊 = 𝐹𝐹𝑑𝑑 cos 𝜃𝜃 𝑃𝑃 = 𝑊𝑊𝑐𝑐

𝑇𝑇𝑇𝑇 = 𝐾𝐾𝑇𝑇 + 𝑃𝑃𝑇𝑇𝑔𝑔 𝐾𝐾𝑇𝑇 = 12𝑚𝑚𝑣𝑣2

𝑃𝑃𝑇𝑇𝑔𝑔 = 𝑚𝑚𝑚𝑚ℎ

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?

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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.

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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)

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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:

Σ𝐹𝐹 = 𝑚𝑚𝑎𝑎

Equation 1: 𝑣𝑣𝑓𝑓 = 𝑣𝑣𝑖𝑖 + 𝑎𝑎𝑡𝑡

Equation 2: ∆𝑥𝑥 = 𝑣𝑣𝑖𝑖𝑡𝑡 + 12𝑎𝑎𝑡𝑡2

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.

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122

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 𝑣𝑣𝑓𝑓 =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).

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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 non-conservative 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.

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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 𝑣𝑣 – tangential velocity 𝑑𝑑 – radius from axis of rotation 𝜔𝜔 – angular velocity/angular frequency 𝜔𝜔𝑖𝑖 –initial angular velocity 𝜔𝜔𝑓𝑓 – final angular velocity 𝑎𝑎 – tangential acceleration 𝛼𝛼 – angular acceleration 𝑎𝑎𝑐𝑐 – centripetal acceleration ∆𝜃𝜃 –angular displacement 𝑡𝑡 – time 𝑇𝑇 – time period

𝑑𝑑 – frequency 𝜏𝜏 – torque F – force 𝜙𝜙 – the angle between the force and radius ectors 𝑑𝑑𝑑𝑑 – force factor/g-force 𝐹𝐹𝑁𝑁 – normal force on you from the seat 𝑚𝑚 – your mass 𝑚𝑚 – acceleration due to gravity Σ𝐹𝐹 – the net force acting on an object 𝑤𝑤 – your weight

PART 1 (Acceleration) Useful Formulae:

𝜃𝜃𝑟𝑟𝑐𝑐𝑐𝑐𝑖𝑖𝑐𝑐𝑟𝑟𝑟𝑟 = 𝜃𝜃𝑐𝑐𝑐𝑐𝑔𝑔𝑟𝑟𝑐𝑐𝑐𝑐𝑟𝑟 �𝜋𝜋

180°� ∆𝜃𝜃 = 1

2�𝜔𝜔𝑖𝑖 + 𝜔𝜔𝑓𝑓�𝑡𝑡

1 𝑑𝑑𝑡𝑡 = 0.3048 𝑚𝑚 𝜔𝜔𝑓𝑓 = 𝜔𝜔𝑖𝑖 + 𝛼𝛼𝑡𝑡

𝑣𝑣 = 𝑑𝑑𝜔𝜔 𝜔𝜔𝑓𝑓2 = 𝜔𝜔𝑖𝑖2 + 2𝛼𝛼Δ𝜃𝜃

𝑎𝑎 = 𝑑𝑑𝛼𝛼 Δ𝜃𝜃 = 𝜔𝜔𝑖𝑖𝑡𝑡 + 12𝛼𝛼𝑡𝑡2

𝑎𝑎𝑐𝑐 = 𝑣𝑣2

𝑟𝑟= 𝑑𝑑𝜔𝜔2

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Use the following diagram to complete the activities in Part 1:

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) 𝚫𝚫θ Δθ (rad) α (rad/s2) ωf (rad/s) v (m/s) ac (m/s2)

Initial direction change (Figure A)

45°

Second direction change (Figure B)

100°

Midway direction change (Figure C)

135°

a. At each of these points, measure the time, 𝑡𝑡, 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, ∆𝜃𝜃, 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).

𝑡𝑡𝑖𝑖 = 0 𝑠𝑠

45°

100° 135°

A B C

𝑣𝑣 = 0 𝑚𝑚/𝑠𝑠

𝑣𝑣 = 0 𝑚𝑚/𝑠𝑠 𝑣𝑣 = 0 𝑚𝑚/𝑠𝑠



𝑡𝑡𝑓𝑓 𝑡𝑡𝑓𝑓 𝑡𝑡𝑓𝑓

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126

c. Use the radian angles that you just calculated in 1b and the measured times in 1a to calculate the angular accelerations, 𝛼𝛼, of either arm during these intervals and record these data in the table above. Show one sample calculation below.

d. Knowing 𝜔𝜔𝑖𝑖 = 0, use the data from the chart to calculate the angular velocity, 𝜔𝜔𝑓𝑓, 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, 𝑣𝑣, and record them in the chart. Show a sample calculation below.

f. Calculate the centripetal acceleration, 𝑎𝑎𝑐𝑐, at each point using the tangential velocities, 𝑣𝑣, or the angular velocities, 𝜔𝜔𝑓𝑓, that were previously found. Record these data in the chart and show a sample calculation below.

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127


1. Were the values for the angular accelerations, 𝛼𝛼, 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)?

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PART 2 (Oscillations) Useful Formulae:

𝑇𝑇 = 1𝑓𝑓

= 2𝜋𝜋𝜔𝜔

𝜏𝜏 = 𝑑𝑑𝐹𝐹 sin𝜙𝜙 𝜔𝜔 = 2𝜋𝜋𝑑𝑑

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, 𝑇𝑇. 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, 𝜔𝜔, 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).

% 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 100 ��𝑇𝑇𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓−𝑇𝑇𝑏𝑏𝑓𝑓𝑏𝑏𝑏𝑏𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓�𝑇𝑇𝑓𝑓𝑎𝑎𝑎𝑎𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎

�

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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, 𝜏𝜏 (m·N) Angular Position, 𝜃𝜃 (radians)

0

−𝜏𝜏𝑚𝑚𝑐𝑐𝑚𝑚

0

𝜏𝜏𝑚𝑚𝑐𝑐𝑚𝑚

0 2π


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?

𝜏𝜏𝑚𝑚𝑐𝑐𝑚𝑚

−𝜏𝜏𝑚𝑚𝑐𝑐𝑚𝑚

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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.

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PART 3 (Force Factor/g-Force) Useful Formulae:

𝑑𝑑𝑑𝑑 = 𝐹𝐹𝑁𝑁𝑚𝑚𝑔𝑔

Σ𝐹𝐹 = 𝑚𝑚𝑎𝑎𝑐𝑐 𝑤𝑤 = 𝑚𝑚𝑚𝑚

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.

𝑑𝑑𝑑𝑑𝑐𝑐𝑡𝑡𝑎𝑎 = 𝑑𝑑𝑑𝑑𝑏𝑏𝑡𝑡𝑐𝑐𝑐𝑐𝑡𝑡𝑚𝑚 =

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.

𝐹𝐹𝑁𝑁𝑡𝑡𝑓𝑓𝑡𝑡 = 𝐹𝐹𝑁𝑁𝑏𝑏𝑓𝑓𝑡𝑡𝑡𝑡𝑓𝑓𝑏𝑏 =

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).

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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.

𝑎𝑎𝑐𝑐𝑡𝑡𝑓𝑓𝑡𝑡 = 𝑎𝑎𝑐𝑐𝑏𝑏𝑓𝑓𝑡𝑡𝑡𝑡𝑓𝑓𝑏𝑏 =


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?

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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.

Documents

Thrill U. · PDF fileThrill U. THE PHYSICS AND MATHEMATICS OF AMUSEMENT PARK RIDES . Physics . ... the dynamics of the rides as a review, incorporate them within a laboratory practical,