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