Conservation of Energy
Energy can be defined as the capacity for doing work. It may exist in a variety of forms
and may be transformed from one type of energy to another. However, these energy
transformations are constrained by a fundamental principle, the Conservation of Energy
principle. One way to state this principle is "Energy can neither be created nor
destroyed". Another approach is to say that the total energy of an isolated system remains
constant even as energy changes from one form to another.
Kinetic Energy:
Ek = ½ mv2
where m = mass, v = velocity
Kinetic energy is energy of motion. The kinetic energy of an object is the energy it
possesses because of its motion. Kinetic energy is an expression of the fact that a moving
object can do work on anything it hits; it quantifies the amount of work the object could
do as a result of its motion
Gravitational Potential Energy: Eg = mgh
where m = mass, g =9.8 m/s2, h = height
Gravitational potential energy is energy an object possesses because of its position in a
gravitational field. The most common use of gravitational potential energy is for an
object near the surface of the Earth where the gravitational acceleration can be assumed
to be constant at about 9.8 m/s2. Since the zero of gravitational potential energy can be
chosen at any point (like the choice of the zero of a coordinate system), the potential
energy at a height h above that point is equal to the work which would be required to lift
the object to that height with no net change in kinetic energy.
Elastic Potential Energy: Ee = 1/2kx2
where k = spring constant, x = change in length of spring
Elastic potential energy is stored as a result of deformation of an elastic object, such as
the stretching of a spring. It is equal to the work done to stretch the spring, which
depends upon the spring constant k (measured in N/m) as well as the distance stretched.
According to Hooke's law (F = kx), the force required to stretch the spring will be
directly proportional to the amount of stretch.
Chemical Potential Energy: Echem
Consider the ability of your body to do work. The glucose (blood sugar) in your body is
said to have "chemical energy" because the glucose releases energy when chemically
reacted (combusted) with oxygen. Your muscles use this energy to generate mechanical
force and also heat. Chemical energy is really a form of microscopic potential energy,
which exists because of the electric and magnetic forces of attraction exerted between the
different parts of each molecule - the same attractive forces involved in thermal
vibrations. These parts get rearranged in chemical reactions, releasing or adding to this
potential energy. Gasoline in a car is another example of chemical potential energy.
Dissipated Energy: Ediss
Consider the case of a moving car that puts on its brakes and comes to a stop. The car
begins by having kinetic energy due to its motion. When it stops, all this kinetic energy
has been transformed to heat – the brakes, road and tires will be warmer. This energy
radiates off into the air and cannot be recovered to do additional work on the car. In this
unit, we will call this dissipated energy.
What is work?
The Work/Energy Principle:
The change in the energy of an object is equal to the net work done on the object.
Modeling Tools for use with Energy including examples
Energy Interactions: All energy interactions can be characterized as energy transfer
mechanisms or energy storage modes, depending on how the system is defined. Energy
storage modes are kinetic, potential and internal energies, designated as ∆E with
corresponding subscripts (∆Ek + ∆Eel + ∆Eg + ∆Eint +∆Echem = ∆E). Energy transfer
mechanisms are working (W), heating (Q), and radiating (R) .
The relationship between energy storage and transfer is shown by the 1st Law of
Thermodynamics, ∆E= W (+ Q + R). This is shown by the system schema below:
It shows that energy transferring into and out of the system affects the nature of the
energy storage in the system. The 1st Law of Thermodynamics and the Law of
Conservation of Energy state that the algebraic sum of these energy changes and transfers
must add up to zero, accounting for all changes relative to the system. This crucial
concept is incorporated into the pie chart and bar graph representational tools used in this
unit.
System Schema: A representation of the system that includes system boundaries,
objects being included in the system being considered and the interactions between these
objects. The system schema represents the first level of abstraction from pictorial
representation, which allows for gentler transitions to further levels of abstraction. Some
of the important features of the system schema are that it defines the system, which has
implications on local energy conservation, as well as representing interactions.
Interactions between objects are represented by two headed arrows and labeled as to the
type of interaction.
Example Situation Statement: A person lifts a book from the ground.
Example System Schema:
Earth
g
c
Person
c
g
c
Book
In this system energy is conserved, since the person is inside the outer dashed line that
defines the system. Objects are represented by boxes in order to remove extraneous
detail from the situation. The interactions labeled in this schema are c = contact, g =
gravitational. Other common interactions include e = electric, m = magnetic, or r =
radiative. In the schema shown above the contact interaction between the book and the
earth is represented by a dashed line indicating this is a time dependant interaction. The
book and earth are not in contact at all times so they get a dashed line. The schema also
forces students to consider which interactions are not critical. No gravitational
interaction between person and book is represented, this shows the interaction is so small
that it can be ignored. The inclusion and exclusion of interactions is based on what are
you trying to model.
Example Situation Statement: A swinging pendulum
One Example System Schema:
In the case of the swinging pendulum, the gravitational interaction between the Earth and
the pendulum bob can be identified as the primary interaction. Note that we assume here
that the problem begins with the total energy of the system determined before hand.
Initially the total energy of the system was set by the "pull-back distance"or height the
pendulum bob was raised from its rest position. What was the source of that energy?
What kind of energy is it? Examining the System Schema makes it easy to identify all
possible interactions that affect the swing cycle of the pendulum. The frictional
interaction between the pivot point in the ceiling and the string tends to slow the
pendulum motion. The interaction of the string tied to the pendulum bob creates a
varying upward tension on the bob. The interaction between the swinging bob and the
surrounding air creates a drag effect on the bob (air resistance) that, although very small,
tends to slow the motion of the bob.
Energy Pie Charts: These are a visual and conceptual representation of the equation of
everything. The total energy in the system is represented by the size of the pie. Energy
transfers into the system are accompanied by an increase in size of the pie and conversely
a transfer out of the system decreases the size of the pie. Pies are also divided according
to the energy storage mechanisms being used, although the divisions are not necessarily
representative of relative amounts of energy. By changing the division of pies as time
progresses, internal energy transfers are represented. Pie Charts are an intermediate
level of abstraction; students are required to look at the energy in the system, but not
concern themselves with the mathematics. They visually emphasize the conservation of
energy, and necessity of definition of system.
Example Situation Statement: A person lifts a book from the ground.
Example of Energy Pie Charts: Using system of person, book & earth.
Echem
Echem Ek
Eg
Eg
Initially
Middle of Lift
Finally
The pie charts shown above show that energy is conserved, because all pies are the same
size. The energy in the person (Echem) decreases throughout the lift, and transfers that
energy to kinetic during the lift, and finally to the gravitational interaction between the
book and the earth. Because the energy at the end is all Eg, we can assume that the
person is 100% efficient in lifting the book since there is no Eint at the end of the lift.
Example Situation Statement: Swinging pendulum.
Example of Energy Pie Charts: Using system of pendulum and earth.
In the case of an idealized swinging pendulum, the total energy is the sum of the
gravitational potential and kinetic energies. At the initial pull back of the pendulum bob,
the total energy is in the form of gravitational potential energy, and kinetic energy is zero,
in the ideal case. When the bob reaches maximum velocity at the bottom of its arc, the
gravitational potential energy is zero, and the total energy is in the form of kinetic energy.
Between these two points (maximum and minimum heights of the bob), the total energy
of the pendulum (the System) is constant, while being continuously transferred back and
forth between potential and kinetic forms of energy. A series of Energy Pie Charts with
the sectors indicating the relative amounts of potential and kinetic energies as a function
of time would summarize the ideal Pendulum problem. For this ideal case (where friction
and air resistance have been ignored) the sizes of the Energy Pie Charts would remain
constant with time, indicating that the total energy of the system was constant. In other
words that energy is conserved, provided all relevant variables in the system had been
correctly identified.
However, in reality we know that the pendulum eventually stops swinging due to friction
at the pivot point in the ceiling and air resistance (minor effect). So the correct way to
draw the Energy Pie Charts (if friction and air resistance were to be ignored in the
problem) would be to indicate energy loss to the System by shrinking the size of each
successive pie chart as shown. Note how the relative amounts of gravitational potential
and kinetic energies change during the swing cycle of the pendulum. Can you identify the
state of the pendulum bob for each of the Energy Pie Charts in the diagram?
Example Situation Statement: Swinging pendulum.
Example of Energy Pie Charts: Using system of pendulum, air, pivot point, earth.
For the more realistic case of the pendulum problem where friction at the pivot point and
air resistance are included in your System Schema, your System would gradually transfer
some of the kinetic energy into heat at the pivot point due to friction, and each successive
pie chart would have the same size (provided all energy sources had been identified in the
problem). Only when energy is conserved (e.g. you have included all forms of energy in
the System) can the series of Energy Pie Charts have the same size as a function of time.
This set of Energy Pie Charts identifies additional energy forms in the Swinging
Pendulum System, and indicates that the total energy of the System identified is
conserved.
Energy Bar Charts: Energy bar charts are a visual and quasi-mathematical
representation of the equation of everything. They are similar to energy pie charts in that
the total height of the bars represents the total energy in the system, and they have
different storage mechanisms that are represented by different bars. The primary
difference between bar charts and pie charts are that bar charts can represent negative
energy. A second, more subtle difference is that the bar charts are more suited to
representing proportions of total energy accurately. Energy transfers into or out of the
system are represented by bars with arrow head either heading into or out of the axes.
Example Situation Statement: A roller coaster cart (ideal situation with no friction).
Example Bar Charts: Using the cart and earth as the system.
Note that the total of Ek + Eg is constant with no energy transferred in or out of the
system.
Example Situation Statement: A spring is stretched.
Example Bar Charts: Using the spring as the system.
Spring begins with no stored elastic energy. Energy is transferred into the spring by the
means of work (an outside force exerted over a distance). The resulting elastic energy
(Ee) is equal to the work done.