Engineering 1182:
Roller Coaster Dynamics-1:
Energy Conservation
Measuring Quantities
Systems of Units
International
System (SI)
English System
(FPS)
Base quantity
Name
Symbol
Name
Symbol
length
meter
m
foot
ft
mass
kilogram
kg
pound
(old:slug)
lb
time
second
s
second
s

For the rest of this lecture, we’ll be using SI
units (metric)
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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).
dS
v

dt
ACCELERATION- Change in velocity per unit time (Units:
m/s2).
2
dv d S
a
 2
dt dt

MASS- A physical property of an object that identifies its
resistance to having a velocity change (Units: kg). m
3
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
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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.
5
Mechanical Energy

When a force, F, is applied to
an object, the energy that is
transferred to the object is
given by E  Fd where d is
the distance over which the
force is applied.
6
Law of Conservation of Energy (COE)


Energy can neither be created nor
destroyed.
Energy can only be changed from one
form to another.
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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.
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Forms of Energy in a Rolling Ball
Energy of the Ball
Potential Energy (PE)
Kinetic Energy (KE)
Total Mechanical Energy of the ball = PE + KE
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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
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Potential Energy


Gravitational Potential Energy
is the energy stored in a body
Mass= 2 kg
due to its height (h). The
PE = ?
height is always measured
relative to some reference
h = 1.52 meters
level (here the ground)
An object of mass m at a
vertical height h above the
ground has a potential PE of the ball shown = mgh
= 2(9.81)(1.52) = 30 Joules
energy of mgh
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Potential Energy
?? If we wanted the
ball to have 75
joules of energy,
what height should
it be raised to?
75J
Mass= 2 kg
h = ? meters
h = PE/(m*g) = 75J / (2kg*(9.81m/s2)) = 3.82 m
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PE Examples



PEA = PEB
PEB > PEC > PED > PEE
PEE = 0 Joules
(Assuming that all the balls
have the same mass)
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Kinetic Energy in a Rolling Ball
Kinetic Energy (KE)
Translational Kinetic
Energy (TKE)
Rotational Kinetic Energy
(RKE)
Kinetic Energy of the ball = TKE + RKE
A rolling ball has both forms of Energy!
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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
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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
16
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)
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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.
18
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 = 0.128 kg-m2
RKE = ½ I ω2 = 250 joules
19
Angular Velocity (ω) vs Linear Velocity (V )
Linear Velocity (V ) 
Change in Linear Displaceme nt ( Meters)
Time( Seconds)
Angular Velocity ( ) 
Change in Anglular Displaceme nt ( Radians )
Time( Seconds)
ω
R
v
V  R
This relationship between linear and angular
velocities holds if and only if the ball is not slipping
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Effective Rolling Radius
Rails
•
•
•
R’
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.
21
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 !
22
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.
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Let’s put it together !

For the ball rolling along the roller coaster
track, between any two subsequent points:
PE1+ TKE 1+ RKE1
= PE2 + TKE2 + RKE2 + “Energy Losses”
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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
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