Modeling Tools for use with Energy including examples
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.
One Example System Schema:
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.
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 definition from above.
The pie charts shown above show that energy is conserved, because all pies are the same size. The energy in the person 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 EIg, we can assume that the person is 100% efficient in lifting the book since there is no Einternal at the end of the lift.
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 meteor falls to earth from outer space.
Example Bar Charts: Using the earth and the meteor as the system.
The total energy in this system is conserved, as can be seen in the bar charts above. The sum of the bars for the different storage mechanisms all add to the same total energy. In each case they add to +1 block of total energy. But because they can show negative, both the kinetic and gravitational interaction energy are growing in magnitude.
Energy vs. distance graphs: The energy pie charts or bar charts show energy of the system as time develops. They can be interpreted also to show energy as a function of distance. Once this interpretation has taken place it is useful to use explicit representations of energy vs. distance. Inherently this representation is for the interaction between two particles: two atoms, a meteor and the earth, etc. Energy vs. distance graphs are used widely in all areas of physics and chemistry, and can be used to explain a wide variety of phenomena such as: cohesion, physical bonding, bound states, change of phase, compressibility, thermal expansion, frictional energy transfers, as well as others.
Example Situation Statement: A meteor falls to earth from outer space and lands on the ground.
Example Energy vs. Distance graph:
There are many variations on the graph shown above. Often only the Interaction Energy would be shown. In this case I have represented Ek and ETotal as well to further elicit the wealth of information that can be extracted from this graph. The above graph shows a number of things, primarily the conservation of energy, which can be seen from the ETotal line being constant. Though the resolution on this picture is not great, you can see the 1/r2 relationship between distance and EI. And as the EI becomes more negative, the Ek must become more positive, so the meteor speeds up on the way down. The meteor in this case is not bound to the earth; once the EI goes to zero, the meteor still has Ek. After the meteor hits the earth you can see there is some compression between the two particles because the graph is sloped, though after hitting the interaction is no longer gravitational. Finally this process must be due to a conservative interaction, because you can not tell whether this is falling or taking off.