Quantifying Energy Conservation

Like many topics in physics, energy can seem like an endless stream of equations to memorize and choose between.  Actually there is one essential equation for energy with two categories of supporting equations.  The essential energy equation is the conservation equation:

\Delta  E_{sys}=  \sum  E_{inputs} - \sum  E_{outputs}       Inputs & Outputs

This equation is often called the first law of thermodynamics and simply says that the change in the total energy of any system is always equal to the total inputs minus the total outputs from that system.  You will see many versions of this equation for special cases (work-kinetic energy theorem, etc...) and it's helpful to remember that they all come from the same equation (Energy and the Confused Student IV: A Global Approach to Energy, Jewitt 2008).  This equation can be expanded to show all of the different ways in which energy can change along with the different mechanisms for energy input and output:

\Delta  E_{k,tran}+ \Delta  E_{k,rot}+ \Delta  E_{th}+ \Delta  E_{grav}+... = W+Q+ T_{MT}+...

On the left side of this equation are changes in energy in all of the various forms it can take.  On the right side are all of the various ways in which energy can transferred to or from a system.

 

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