Mechanical Energy
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The sum of the kinetic energy and any potential energies of a system. |
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:
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| Wiki Markup |
{excerpt}The sum of the [kinetic energy] and any [potential energies|potential energy] of a system.{excerpt} ||Page Contents|| |{toc:indent=10px}| h2. 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|gravity#negpe], and [elastic potential energy|Hooke's Law#epe]. Collectively, these three types of energy are classified as mechanical energy because of their role in the mechanics of macroscopic bodies. h2. 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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(
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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:
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}\begin{large}\[ E = K + U_{g} + U_{e}\]\end{large}{latex} h2. 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}{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: |
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:
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{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 for Mechanical Energy 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. Since gravitation (universal) 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.
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When a system is sliding along a (non-accelerating) surface, it is possible to include a non-conservative normal force (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 normal force from the surface. Thus, the dot product of the normal force with the path will always be zero and the normal force will contribute zero work. Since [gravity] and [spring forces|Hooke's Law] 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. h2. Examples of Systems Conserving Mechanical Energy h4. Non-Conservative Forces Absent One clear way to ensure that the work done by forces other than gravity and springs is zero is to design a system that exeperiences no other forces. Thus, for example, any system which involves pure freefall, or freefall after launch from a spring, or freefall onto a spring, etc, will clearly conserve mechanical energy (see figures below). h4. Movement Along Frictionless Surface When a system is sliding along a (non-accelerating) surface, it is possible to include a normal force (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 normal force from the surface. Thus, the dot product of the normal force with the path will always be zero and the normal force will contribute zero work. Thus, assuming frictionless surfaces, systems like those sketched below will be compatible with mechanical energy conservation. {note}Note that the presence of a friction force _will_ disrupt conservation of mechanical energy, since the dot product of the friction with the velocity of the object will always be nonzero and negative.{note} |