How to Create and Interpret Energy Diagrams

Roshni Desai, Fall 2023

The Main Idea

Energy diagrams are tools used to analyze a system's energy and motion with respect to a scalar variable like position or time. They are typically used to represent the kinetic and potential energy within a system, in addition to a horizontal line that depicts the total mechanical energy of the system.

To draw the energy graph of a system, the following method should be used:

1. Determine if the potential energy is attractive or repulsive
   • For example, gravitational potential energy is attractive since it draws objects to the surface of the Earth [math]\displaystyle{ \left(U_g \lt 0\right) }[/math].
   • Electric potential energy for charges with the same sign is repulsive, since like charges repel [math]\displaystyle{ \left(U_e \gt 0\right) }[/math].
2. Analyze whether the system is bound, unbound, or at escape speed to determine the location of the total energy line
   • Bound System: A system in which the total energy is negative
     • [math]\displaystyle{ E = K + U \lt 0 }[/math]; horizontal line is below the x-axis
     • The distance between the objects in the system is limited and quantifiable
   • Unbound System: A system in which the total energy is positive
     • [math]\displaystyle{ E = K + U \gt 0 }[/math]; horizontal line is above the x-axis
     • If [math]\displaystyle{ r }[/math] approaches [math]\displaystyle{ \infty }[/math], the distance between the objects in the system is infinite and unquantifiable; the kinetic energy cannot equal 0
   • System at Escape Speed: A system in which the total energy is equal to 0
     • [math]\displaystyle{ E = K + U = 0 }[/math]; horizontal line is on the x-axis
3. Draw the kinetic energy line/curve – this is always positive!
   • This is usually the reverse of the potential energy curve because [math]\displaystyle{ K + U = E }[/math]

A Mathematical Model

The mathematical model derived from energy graphs comes down to the fundamental principle: [math]\displaystyle{ E = K + U }[/math], where [math]\displaystyle{ E }[/math] is the total energy, [math]\displaystyle{ K }[/math] is the kinetic energy, and [math]\displaystyle{ U }[/math] is the potential energy (gravitational, electric, spring, etc.) of the system. This model can be modified, however, depending on the type of system:

• Bound System: A system in which the total energy is negative
  • [math]\displaystyle{ E = K + U \lt 0 }[/math]; horizontal line is below the x-axis
  • The distance between the objects in the system is limited and quantifiable
• Unbound System: A system in which the total energy is positive
  • [math]\displaystyle{ E = K + U \gt 0 }[/math]; horizontal line is above the x-axis
  • If [math]\displaystyle{ r }[/math] approaches [math]\displaystyle{ \infty }[/math], the distance between the objects in the system is infinite and unquantifiable; the kinetic energy cannot equal 0
• System at Escape Speed: A system in which the total energy is equal to 0
  • [math]\displaystyle{ E = K + U = 0 }[/math]; horizontal line is on the x-axis

A Computational Model

Vpython is great for modeling this concept. Using vpython, we can model many different systems that have kinetic and potential energy. We can model a spacecraft orbiting the Earth, and we can create graphs to display the kinetic, potential, and kinetic+potential energies of this system. See this code for how to do this!

[Sample Vpython code:https://trinket.io/glowscript/4010e21bc3]

Examples

Difficult

creating energy graphs for different situations

Connectedness

One of my strongest passions is sustainability and conservation, particularly renewable energy systems. Energy diagrams, as we have learned, provide information about the kinetic and potential energies of a system. For example, the potential energy within a dam could be determined from the elevation of water, while the kinetic energy of the water could be found by analyzing its speed due to currents and turbines. This also ties into how energy graphs connect with my major, Materials Science and Engineering. Although this field has several paths, one route that I am interested in is how materials can be engineered to prevent or solve environmental problems. As a result, energy diagrams are beneficial in many industrial applications, including optimizing processes to control pollution and developing methods to minimize both household and industrial waste. Apart from environmental engineering, though, energy diagrams can be used to in electronics and semiconductors as well and are important for understanding how to design electronical equipment and enhancing performance of batteries, which contain electric potential energy that allows for the kinetic flow of electrons. Furthermore, energy diagrams are used to analyze on a smaller molecular scale as well, by representing changes in chemical reactions and catalytic processes.

History

The concept of mechanical energy was not solely discovered by a single person at a single time. It was developed as a result of work done by scientists over several centuries of research, after which it was able to be represented graphically as a combination of all the types of energies contained within a system.

See also

Here are some interactive simulations to that may help with understanding representation of energy conservation!

1. PhET simulation to understand the conservation of energy within a system
   • https://phet.colorado.edu/en/simulations/energy-skate-park
2. PhET simulation to obtain a better understand of energy conservation in springs
   • https://phet.colorado.edu/en/simulations/masses-and-springs/about

Further reading

1. Article to learn more about the Law of Conservation of Energy
   • https://byjus.com/physics/law-of-conservation-of-energy/
2. Article to highlight important observations about energy diagrams and how to interpret them
   • https://www.dummies.com/article/academics-the-arts/science/physics/how-to-interpret-energy-diagrams-in-physics-143033/

External links

Video explaining how to read spring energy graphs: https://www.youtube.com/watch?v=HuV8itFrht4 Video discussing the concept of mechanical energy: https://www.youtube.com/watch?v=VrLVUfB_t_U

References

https://phys.libretexts.org/Courses/University_of_California_Davis/UCD:_Physics_9HA__Classical_Mechanics/3:_Work_and_Energy/3.7:_Energy_Diagrams https://www.dummies.com/article/academics-the-arts/science/quantum-physics/measuring-the-energy-of-bound-and-unbound-particles-161223/