The sum of the kinetic energy and any potential energies of a system.
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Motivation for Concept
It is a fundamental postulate of physics that energy is not created or destroyed. The total energy of the universe is fixed, but it can be transformed from one type of energy to another. In order to make wide use of this principle, many types of energy must be considered (e.g. nuclear, chemical/electrostatic, gravitational, kinetic, etc.). To give a first illustration of the principle, however, introductory physics introduces three energy types that are related to the topics of mechanics: kinetic energy, [gravitational potential energy], and [elastic potential energy]. Collectively, these three types of energy are classified as mechanical energy because of their role in the mechanics of macroscopic bodies.
Mathematical Definition of Mechanical Energy
The mechanical energy (E) of a system is the sum of the system's kinetic and potential energies:
\begin
[ E = K + U]\end
In introductory mechanics, it is basically assumed that the possible constituents of the potential energy are gravitational and elastic potential energy, so that the mechanical energy is essentially:
\begin
[ E = K + U_
+ U_
]\end
Generalized Work-Energy Theorem
The Work-Kinetic Energy Theorem states:
\begin
[ \Delta K = W_
]\end
where Wnet is the total work from all sources done on a point particle system.
We can now generalize this theorem slightly to allow for a system composed of rigid bodies that have only mechanical interactions (they do not emit radiation, transfer heat, etc.). We allow for the system to have conservative interactions, but we remove these interactions from the net work and instead account for them using potential energy. Essentially, we move the contribution of conservative forces from the right hand side of the Theorem to the left hand side. The result is:
\begin
[ \Delta E = W^
_
]\end
so that the right hand side is the sum of the works arising from all forces that do not have an associated potential energy.
Mechanical Energy Conservation
Conditions for Mechanical Energy Conservation
From the generalized Work-Energy Theorem, we see that the mechanical energy will be constant (assuming only mechanical interactions) when the net non-conservative work done on the system is zero. Since [gravity] and [spring forces] are the only conservative forces commonly treated in introductory mechanics, this condition usually amounts to the constraint that the total work done by forces other than gravity or spring forces is zero.