Engineering 1182:

Roller Coaster Dynamics-1: Energy Conservation

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Measuring QuantitiesSystems 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).

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

dt

dSv

2

2

dt

Sd

dt

dva

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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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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.

Physics Concept - Energy

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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.

dFE

Mechanical Energy

d

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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

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.

Main Elements of Roller Coaster System

= Ball + Rails + Structure

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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

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KEinitial + PEinitial = KEfinal + PEfinal

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)

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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 kgPE = ?

PE of the ball shown = mgh = 2(9.81)(1.52) = 30 Joules

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Potential Energy

?? If we wanted the ball to have 75 joules of energy, what height should it be raised to?

h = ? meters

Mass= 2 kg75J

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

Translational Kinetic Energy (TKE)

Kinetic Energy (KE)

Rotational Kinetic Energy (RKE)

Kinetic Energy of the ball = TKE + RKEA 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

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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.

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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

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Angular Velocity (ω) vs Linear Velocity (V )

)(

)()(

SecondsTime

MetersntDisplacemeLinearinChangeVVelocityLinear

)(

)()(

SecondsTime

RadiansntDisplacemeAnglularinChangeVelocityAngular

RV

This relationship between linear and angular velocities holds if and only if the ball is not slipping

ω

v

R

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Effective Rolling Radius

R’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.

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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 !

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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.

Energy Transfers (continued)

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Let’s put it together !

For the ball rolling along the roller coaster track, between any two subsequent points:

+ +

= + + + “Energy Losses”1PE 1TKE 1RKE

2PE 2TKE 2RKE

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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