How to Create and Interpret Energy Diagrams: Difference between revisions

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===A Mathematical Model===
===A Mathematical Model===


The mathematical model derived from energy graphs comes down to the fundamental principle: <math>E = K + U</math>, where <math>E</math> is the total energy, <math>K</math> is the kinetic energy, and <math>U</math> is the potential energy (gravitational, electric, spring, etc.) of the system.
The mathematical model derived from energy graphs comes down to the fundamental principle: <math>E = K + U</math>, where <math>E</math> is the total energy, <math>K</math> is the kinetic energy, and <math>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>E = K + U < 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>E = K + U > 0</math>; horizontal line is above the x-axis
#** If <math>r</math> approaches <math>\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>E = K + U = 0</math>; horizontal line is on the x-axis


===A Computational Model===
===A Computational Model===

Revision as of 23:57, 2 December 2023

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

Simple

Intermediate

Difficult

creating energy graphs for different situations

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