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Mechanical
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Energy
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The |
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sum |
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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.
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nuclear,
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chemical/electrostatic,
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gravitational,
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kinetic,
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etc.).
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To
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give
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a
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first
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illustration
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of
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the
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principle,
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however,
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introductory
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physics
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introduces
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three
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energy
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types
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that
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are
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related
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to
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the
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topics
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of
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mechanics:
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,
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,
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and
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.
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Collectively,
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these
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three
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types
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of
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energy
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are
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classified
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as
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mechanical
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energy
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because
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of
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their
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role
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in
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the
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mechanics
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of
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macroscopic
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bodies.
Mathematical Definition of Mechanical Energy
The mechanical energy (E) of a system is the sum of the system's kinetic and potential energies:
| Latex |
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h3. Mathematical Definition of Mechanical Energy The mechanical energy (_E_) of a system is the sum of the system's kinetic and potential energies: {latex}\begin{large}\[ E = K + U\]\end{large}{latex} |
In
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introductory
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mechanics,
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it
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is
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basically
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assumed
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that
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the
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possible
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constituents
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of
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the
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potential
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energy
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are
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(
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U
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g)and
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...
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(
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U
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e),
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so
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that
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the
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mechanical
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energy
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is
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essentially:
| Latex |
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}\begin{large}\[ E = K + U_{g} + U_{e}\]\end{large}{latex} h3. Generalized |
Generalized Work-Energy
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Theorem
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The
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states:
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}\begin{large}\[ \Delta K = W_{\rm net}\]\end{large} |
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:
| Latex |
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{latex} where _W_~net~ 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: {latex}\begin{large}\[ \Delta E = W^{NC}_{net}\]\end{large}{latex} |
so
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that
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the
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right
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hand
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side
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is
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the
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sum
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of
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the
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works
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arising
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from
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all
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forces
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that
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do
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not
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have
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an
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associated
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potential
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energy.
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Conditions
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for
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Mechanical
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Energy
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Conservation
From the generalized Work-Energy
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Theorem,
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we
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see
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that
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the
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mechanical
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energy
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will
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be
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constant
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(assuming
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only
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mechanical
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interactions)
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when
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the
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net
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non-conservative
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work
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done
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on
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the
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system
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is
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zero.
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Since
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and
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are
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the
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only
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conservative
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forces
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commonly
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treated
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in
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introductory
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mechanics,
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this
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condition
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usually
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amounts
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to
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the
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constraint
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that
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the
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total
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work
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done
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by
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forces
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other
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than
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gravity
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or
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spring
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forces
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is
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zero.
| Note |
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When a system is sliding along a {note}When a [system] is sliding along a(non-accelerating) surface, it is possible to include a _non-conservative _ [ ](in addition to springs and gravity) on the system without changing the mechanical energy. The reason is that an object moving along a surface will always be moving in a direction perpendicular to the [ ]from the surface. Thus, the [ ]of the [ ]with the path will always be zero and the [ ]will contribute zero []. {note} {table-cell} {table-row} {table} {live-template:RELATE license} |