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Mechanical
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Energy
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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:
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{excerpt} h3. {toggle-cloak:id=mot} Motivation for Concept {cloak:id=mot} 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|gravitation#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. {cloak} h3. {toggle-cloak:id=math} Mathematical Definition of Mechanical Energy {cloak:id=math} 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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U
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g)and
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essentially:
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}\begin{large}\[ E = K + U_{g} + U_{e}\]\end{large}{latex} {cloak} h3. {toggle-cloak:id=gen} Generalized |
Generalized Work-Energy
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Theorem
The Work-Kinetic
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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:
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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} |
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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 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. {cloak} h3. {toggle-cloak:id=cond} Conditions for Mechanical Energy Conservation {cloak:id=cond} h4. {toggle-cloak:id=gencond} General Condition {cloak:id=gencond} 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 [gravitation] 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. {cloak:gencond} h4. {toggle-cloak:id=exncabs} Examples: Non-Conservative Forces Absent {cloak:id=exncabs} 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). |!mechecons1.png|valign=bottom!|!mechecons2.png|valign=bottom!|!mechecons3.png|valign=bottom!| {cloak:exncabs} h4. {toggle-cloak:id=exfls} Examples: Movement Along Frictionless Surface {cloak:id=exfls} 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. !mechemoreex.png! {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} {cloak:exfls} {cloak:cond} h3. {toggle-cloak:id=diags} Diagrams and Mechanical Energy {cloak:id=diags} h4. {toggle-cloak:id=isfsdiag} Initial-State Final-State Diagrams {cloak:id=isfsdiag} 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. {cloak:isfsdiag} h4. {toggle-cloak:id=exvl} Example -- Vertical Launch {cloak:id=exvl} 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: |!mecheex1i.png!|!mecheex1f.png!| ||Initial State||Final State|| The act of drawing the diagram will often clarify the givens, and it will also remind us to choose a zero height for the gravitational potential energy. {cloak:exvl} h4. {toggle-cloak:id=exebg} Energy Bar Graphs {cloak:id=exebg} Drawing and labeling the initial-state final-state diagram will give a good idea of where the energy is in the system. Thus, once the system is depicted physically, it is often useful to represent the energy distribution graphically as well. This is done using a bar graph that contains one bar for each distinct type of energy that will make up the mechanical energy of the system. This bar graph is not necessarily quantitative, but it should be drawn to reflect conservation of energy if the non-conservative work is zero. {cloak:exebg} h4. {toggle-cloak:id=exvlc} Example -- Vertical Launch Continued {cloak:id=exvlc} Continuing with the example from above, we can see that in the initial state the energy is divided among spring potential and gravitational potential, with zero kinetic energy present. In the final state, the energy is divided among gravitational potential and kinetic energy, with the spring (returned to its natural position) contributing zero energy. A possible bar graph representation of this situation is shown below: |!mechebar1i.png!|!mechebar1f.png!| ||Initial State||Final State|| Note that these bar graphs are scaled to show conservation of energy. The sum of the three bars should be the same in each case. We have made a guess, however, about the relative sizes of the kinetic and gravitational potential energies. We cannot know which will be larger until we finish solving the problem. We do know, however, that they add up to equal the initial mechanical energy. Thus, a second reasonable guess would have been: |!mechebar1i.png!|!mechebar2f.png!| ||Initial State||Final State|| Either is acceptable as an initial guess at the relative energies. {cloak:exvlc} h4. {toggle-cloak:id=exhtb} Example -- Hit the Brakes {cloak:id=exhtb} Initial-state final-state diagrams and energy bar graphs are also useful for detecting problems in which mechanical energy is _not_ conserved. Consider the following example: A 1500 kg roller coaster car is moving at a speed of 10 m/s moving with no friction when it crests the final hill at a height of 15 m above the end of the ride. It then descends to a level area of track that extends for 20 m before the finish. As soon as the car reaches the level track, it hits the brakes, coming to a stop exactly at the finish line. Determine the effective coefficient of friction experienced by the car during braking. Here, the initial-state final-state diagram and energy bar graphs will be: |!mecheex2i.png!|!mecheex2f.png!| ||Initial State||Final State|| With our choices of coordinates, it is clear that the mechanical energy is _not_ conserved in this problem. Thus, we look for a source of non-conservative work. This gives us a way to attack the friction force, which is the source of non-conservative work in this case. {cloak:exhtb} {cloak:diags} h3. {toggle-cloak:id=relex} Relevant Examples {cloak:id=relex} {contentbylabel:constant_energy,non-conservative_work|maxResults=50|operator=OR|showSpace=false|excerpt=true} {cloak} {td} {tr} {table} {live-template:RELATE license} |