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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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 |
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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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. 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. h4. Examples: 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. Examples: 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} h2. Diagrams and Mechanical Energy h4. Initial-State Final-State Diagrams Because the Work-Energy Theorem and the principle Law of Change for the [Mechanical Energy and Non-Conservative Work] model involve only the initial and final energies of the system, it is useful to devote considerable attention to understanding the system's configuration at those times. It is customary to sketch the system in its initial and final configurations, labeling the quantities that are relevant for the kinetic and potential energies of the system. h4. Example -- Vertical Launch As an example of an initial-state final-state diagram, suppose we were asked to determine the speed of a 0.25 kg block launched vertically from rest by a spring of spring constant 500 N/m that was initially compressed 0.10 m from its natural length when the block reaches a height of 0.50 m above the natural position of the spring. We can neatly summarize these givens through a two-panel diagram: |