1. Roller Coaster Dynamics-1: Energy Conservation
Engineering 1182:
Roller Coaster Dynamics-1:
Energy Conservation

2. International System (SI)
Measuring Quantities
Systems of Units
International System (SI)
English System (FPS)
Base quantity
Name
Symbol
length
meter m
foot ft
mass
kilogram kg
pound (old:slug) lb
time
second s
Stress on the importance of Units as they will be seeing a mixture of SI Units and English Units. English
Units are still common in industry in the USA and so students should expect to see a mix of units after graduating.
SI (Système International d’Unités) is the official designation for the meter-kilogram-second metric system. It is
sometimes called MKS based on the first letters of the primary units. The English system is sometimes called FPS from
the first letters of foot-pound-second. You will also see an alternate metric system called CGS for centimeters-grams-seconds.
For the rest of this lecture, we’ll be using SI
units (metric)

3. Physics Concepts - Definitions
DISPLACEMENT- A measure of HOW FAR and in WHAT DIRECTION an object has MOVED relative to a “starting” point (Units: m). S
VELOCITY- Change in displacement per unit time (Units: m/s).
ACCELERATION- Change in velocity per unit time (Units: m/s2).
MASS- A physical property of an object that identifies its resistance to having a velocity change (Units: kg). m
Velocity describes how quickly an object changes its position. The higher the velocity the quicker an object travels between 2 locations. Phrases like, how fast or how quickly, are used to describe velocity. Often the word speed is substituted for the word velocity in common usage. However, technically the two are different. Velocity is actually speed with direction. For example, 60 mph west is a velocity. "West" is the direction and 60 mph is the speed.
We are intentionally trying to avoid teaching them vector concepts as this will complicate the physics involved. The approach of using Energy equations to design the roller coaster does not need these vector concepts.
Acceleration: Acceleration describes how quickly an object changes its velocity. Phrases like, slow down, speed up, change speed and change velocity are used to describe accelerations. If students want an easy way to determine if they are visualizing acceleration or a constant velocity along a straight line they only need to ask one question, "Is the object slowing down or speeding up?" If the answer is "Yes," then it is accelerating (decelerating). If the answer is "No" then it is moving with a constant velocity.
Changes in acceleration greatly contribute to the thrill of a roller coaster ride. A rider may feel greater sensations in a low-speed coaster with sharp acceleration changes than on a faster coaster with a smoother ride. This is important because the magnitude of accelerations are limited by the physical limitations of typical riders. Pure speed is often not as recognizable as the surge of acceleration during a coaster ride.
Mass can be defined either inertially as above or in some sense as “amount of stuff”. There is no theoretical reason that these two
definitions HAVE to be the same, but nobody has yet found any difference. Physics groups are working on this and on alternate
mass definitions. In practice, mass is defined by comparison with a known standard mass (amount of stuff) by using something like
a scale (equal resistance to having a velocity change under the force of gravity).

4. Physics Concepts - Definitions
FORCE is a “PUSH” or a “PULL” that is defined by its effect on a mass (Units: Newtons, N).
F=ma (1 Newton = 1 kg-m/s2)
WEIGHT- The force acting on a mass when it is subjected to gravity.
F=mg
Where g is the acceleration due to the Earth’s gravitational force
For Standard Gravity use g = 9.81 m/sec2
Example of the use of force: “I applied 450 N force to push my car out of the garage”
Sir Isaac Newton is famous for his work on Gravity. He discovered that gravitational force acts equally on all objects.
The gravitational pull on the surface of the moon is 1/6 that of the earth because the moon has a different mass and different diameter
than the earth
The action of gravity causes an object to be pulled towards the earth
At standard earth gravity, the force of gravity acting on a mass of 1kg is 9.8 N
Weight is a measure of earths gravitational pull on an object. An object can only have weight if gravity is at work
In space, an object will have the same mass as on the surface of the earth but little or no weight - because there is little or no gravity!
In contrast, when skydiving, you have the impression of being weightless because there is little opposing force but you still have weight and are initially accelerating due to the force of gravity.

5. Physics Concept - Energy
ENERGY is a conserved property of an object that relates to its ability to do work. Energy can have a number of forms, for example mechanical, electrical, chemical, or nuclear. E
Units: Joules or N-m (Newton-meter).
There are different formulas describing different forms of energy.
I’m trying to avoid the circular definition that Energy IS the ability to do “Work”, while “Work” is the transfer of Energy from one system to another or from one form to another. PE converts to KE by means of work done by gravity. Work and energy are different in that Work is a process while energy is a property or attribute, but for our purposes, we can talk about energy transfers and not specifically introduce the concept of work. A system’s energy is the ability to produce work, regardless of whether any work actually happens. A raised boulder has energy even though it’s tied up and left that way and nothing ever happens.
Whenever possible, the concepts learned should be tied to the Roller Coaster.

6. Mechanical Energy
When a force, F, is applied to an object, the energy that is transferred to the object is given by where is the distance over which the force is applied.
Mechanical Energy is what we’ll be concerned with during the roller coaster design.
Ex. Energy is used in lifting a book from the table (a force is needed to overcome gravity)
Ex. Energy is used in dragging an object along the floor (a force is needed to overcome friction).

7. Law of Conservation of Energy (COE)
Energy can neither be created nor destroyed.
Energy can only be changed from one form to another.
In Physics, the word ‘conservation’ means that it does not change with time. Energy could certainly change form,(e.g. from P.E to K.E), but the overall energy of the system remains conserved.

8. For A Roller Coaster Main Elements of Roller Coaster System
= Ball + Rails + Structure
For our roller coaster we will represent the cars by a rolling ball.
We only care about the energy stored in the rolling ball. This is only part of the energy of the complete system.
It is very important to talk about “whose” energy we are referring to. Is it the energy of the ball or the energy of the complete roller coaster or the energy contained in the room with the roller coaster?
The ball by itself is a system.
The track by itself is a system.
The structure by itself is a system.
When all these are put together, it forms a bigger system that is different from the individual systems.
Main Elements of Roller Coaster System = Ball + Rails + Structure
When we speak about energy, we need to know which system we are referring to! The law of Conservation of Energy holds true for any closed system, but our roller coaster is NOT a closed system since energy will be transferred from the ball to the surrounding environment. We are only interested in the magnitude of these “Losses” so that we can keep track of the energy of our ball.

9. Forms of Energy in a Rolling Ball
Potential Energy (PE)
Energy of the Ball
Kinetic Energy (KE)
Total Mechanical Energy of the ball = PE + KE
This is a very important breakdown of the various forms of energy associated with a rolling ball. Each form of energy will be discussed in detail in the following slides.

10. Energy Conservation (no friction!)
At the top of a hill, the cars in a roller coaster possess a large quantity of potential energy.
During the first drop, the cars lose much of their potential energy and consequently gain kinetic energy.
Each change in height corresponds to a change of speed as potential energy (due to height) is transformed to and from kinetic energy (due to speed)
KEinitial + PEinitial = KEfinal + PEfinal
TME is Total Mechanical Energy
As the ride continues, the train of cars is continuously losing and gaining height. Each gain in height corresponds to a loss of speed as kinetic energy (due to speed) is transformed into potential energy (due to height). Each loss in height corresponds to a gain of speed as potential energy (due to height) is transformed into kinetic energy (due to speed). This transformation of mechanical energy from the form of potential to the form of kinetic and vice versa is illustrated in the animation.
TMEinitial = KEinitial + PEinitial = KEfinal + Pefinal = TMEfinal
K.E.(Joules) = ½ * mass(kg) * Velocity2[(m/s)2]
P.E.(Joules) = mass(kg) * g(m/s2) * height(m)

11. Potential Energy
Gravitational Potential Energy is the energy stored in a body due to its height (h). The height is always measured relative to some reference level (here the ground)
An object of mass m at a vertical height h above the ground has a potential
energy of mgh
h = 1.52 meters
Mass= 2 kg
PE = ?
Gravitational Potential Energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the heavy ram of a pile driver is dependent on two variables - the mass of the ram and the height to which it is raised.
There is a direct relation between gravitational potential energy and the mass of an object; more massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object; the higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation: GPE=mgh
PE of the ball shown = mgh
= 2(9.81)(1.52) = 30 Joules

12. Potential Energy
h = ? meters
Mass= 2 kg
75J
?? If we wanted the ball to have 75 joules of energy, what height should it be raised to?
It would be great if the students could solve the problem in class. Not only does it provide a way to work on the concepts learned, it also gives a short break from the lecture!
Solution: PE = mgh, so h = PE/mg = 75J/(2kg*9.8m/s2) = 3.82 m
h = PE/(m*g) = 75J / (2kg*(9.81m/s2)) = 3.82 m

13. PE Examples PEA = PEB PEB > PEC > PED > PEE PEE = 0 Joules
Gravitational PE is normally referred to as PE unless otherwise stated.
This slide shows the effect of “height” on the PE of the ball.
This slide assumes that the point E is on the ground (the ground is the reference level), i.e. height is zero and hence no PE.
On the roller coaster, the higher the ball’s starting point, the greater the initial PE of the ball.
(Assuming that all the balls
have the same mass)

14. Kinetic Energy in a Rolling Ball
Translational Kinetic Energy (TKE)
Kinetic Energy (KE)
Rotational Kinetic Energy (RKE)
Kinetic Energy of the ball = TKE + RKE
A rolling ball has both forms of Energy!
This is a very important breakdown of the various forms of kinetic energy associated with a rolling ball. Each form of energy will be discussed in detail in the following slides.

15. Translational Kinetic Energy
An object has Translational Kinetic Energy (TKE) when it is undergoing linear displacement
TKE = ½mv2
m = mass of object
v = velocity of object
The equation reveals that TKE of an object is directly proportional to the square of its speed. That means that for a twofold increase in speed, the kinetic energy will increase by a factor of four. Stress to students that velocities should be measured accurately during experiments to get better results.

16. Translational Kinetic Energy- Example
A 50 gram ball is moving in a straight line with a velocity v= 20 m/s. What is it’s TKE ?
Watch out for the Units!
TKE = ½mv2 = ½(50 x10-3 )(20)2
= 10 Joules
A sample problem for students.

17. Rotational Kinetic Energy (RKE)
An object spinning about an axis is said to have Rotational Kinetic Energy.
RKE = ½Iω2
I: Moment of Inertia
ω: Angular Velocity (radians/sec)
The roller coaster ball will have RKE.

18. Moment of Inertia (I) The moment of Inertia (I) of an object
Measures the resistance an object has to rotating about a particular axis, similar to the way that mass is the object’s resistance to changing its velocity.
Depends on its mass, shape and axis of rotation.
Larger the mass of the ball, more the energy it requires to roll.

19. Rotational KE – Example
A solid sphere of radius 0.4 m and weighing 2kg is rolling with an angular velocity of 62.5 radians/s. Find its Rotational KE.
I = (2/5) x M x R2 = kg-m2
RKE = ½ I ω2 = 250 joules
Sample problem to calculate the RKE of a ball.

20. Angular Velocity (ω) vs Linear Velocity (V )
This slide is only to introduce them to Angular Velocity in terms of Linear velocity. This relation is enough for the purposes of our course.
Linear Displacement is the DISTANCE through which the object MOVES (linearly).
Angular displacement is the ANGLE through which the object TURNS.
For completeness, V=ωR only if no slippage occurs.
This relationship between linear and angular
velocities holds if and only if the ball is not slipping

21. Effective Rolling Radius
Rails
The ball sits down between the tracks making the rolling radius smaller.
The angular velocity is increased.
If the rails are not supported and split further apart, the ball will sit farther down.
Picture shows a cross-section of the ball sitting on the rails. Due to the nature of contact between the ball and the rails, the geometric radius of the ball cannot be used in our calculations. Instead, we define “rolling radius” as shown in the picture. To first order, this effect is included in the design spreadsheet discussed in Basics. The distance between track supports is one of the trade-offs that students need to consider when designing their roller coasters.

22. Energy Transfers
As the ball rolls down the roller coaster track, some energy of the moving ball is:
Lost to friction and dissipated as heat
Spent in overcoming Air Resistance
Lost to Structural Deformation
Converted to Sound Energy
Unwanted
Energy
Losses !
Energy transferred away from the ball is sometimes called Energy Losses because the energy is lost from what we’re interested in (in this case the ball).

23. Energy Transfers (continued)
In general, energy transferred away from the ball will NOT come back, and so the total mechanical energy of the ball will be always decreasing.
In the real world, we cannot avoid losses but can only MINIMIZE and/or ALLOW for them.
While it’s possible to conceive of air currents pushing the ball and adding some energy, in an environment like the lab rooms any “gain” will be negligible and should be ignored.

24. Let’s put it together ! + + = + + + “Energy Losses”
For the ball rolling along the roller coaster track, between any two subsequent points:
= “Energy Losses”
This slide should provide a summary of the concepts they learned and how it all fits into the big picture- The Roller Coaster.
Losses can be referred to if there is motion from one point to another point. The energy of the ball is always PE+TKE+RKE for our purposes. If we compare energy at two or more different locations of the ball, then we can talk of ‘gains’ or ‘losses’. In our case, since there is no chain lift or other means of adding energy, the Total Mechanical Energy of the ball at the beginning is equal to the Total Mechanical Energy of the ball at any subsequent point plus “Losses”.

25. Design Considerations
You will be estimating the velocity of the ball at selected points along your roller coaster track using energy calculations to:
Make sure the velocity into turns is not too high (making banking difficult)
Make sure that the ball can reach the top of vertical loops
Make sure that the ball will not fly off the top of bumps