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System: Any system that does not undergo significant changes in internal energy. — Interactions: Any interactions that can be parameterized as mechanical work. Notable exceptions include heat transfer or radiation.

Introduction to the Model

Description and Assumptions

If we ignore non-mechanical processes like heat transfer, radiative losses, etc., then we arrive at a model involving only mechanical energy which changes due to the application (or extraction) of the work done by non-conservative forces The non-conservative forces can be external forces exerted on the system or internal forces resulting from the interactions between the elements inside the system.

Learning Objectives

Students will be assumed to understand this model who can:

Compute the translational kinetic energy of an object.
Compute the rotational kinetic energy of a rigid body rotating about an axis.
Apply the constraint of rolling without slipping.
Define the term non-conservative.
Calculate the work done by a force acting on a moving object.
State the Work-Kinetic Energy Theorem.
Name the conservative forces commonly encountered in mechanics problems.
Explain why the zero point of the (near-earth) gravitational potential energy is arbitrary.
Define the variables appearing in the expression for elastic potential energy.
Calculate the total mechanical energy of a system containing any number of rotating and translating rigid bodies near the surface of the earth that interact via springs.
Construct intitial-state final-state diagrams to summarize the mechanical energy of a system.
Describe the conditions under which mechanical energy is conserved.

Relevant Definitions

Section

Column

Mechanical Energy

Latex
\begin{large}\[E = K + U\]\end{large}

Column

Kinetic Energy

Latex
\begin{large}\[ K = \frac{1}{2}mv^

Wiki Markup

h2. Keys to Applicability Can be applied to any system for which the change in [mechanical energy] can be attributed to [work|work] done by [non conservative forces|non-conservative] (as opposed to processes like heat transfer, radiative losses, etc.). The non-conservative forces can be external forces exerted on the system or internal forces resulting from the interactions between the elements inside the system. It is specially useful for systems where the non-conservative work is zero, in which case the [mechanical energy] of the system is constant. h2. Description {table:align=right|cellspacing=0|cellpadding=1|border=1|frame=box|width=40%} {tr} {td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]* {td} {tr} {tr} {td} {pagetree:root=Model Hierarchy|reverse=true} {td} {tr} {table} || Page Contents || | {toc:style=none|indent=10px} | ---- h2. Assumed Knowledge h4. Prior Models * [Point Particle Dynamics] h4. Vocabulary * [system] * [internal force] * [external force] * [conservative force] * [non-conservative] * [kinetic energy] * [gravitational potential energy] * [elastic potential energy] * [mechanical energy] ---- h2. Model Specification h4. System Structure *[Constituents|system constituent]:*  One or more [point particles|point particle] or [rigid bodies|rigid body]. Technically, the system must be defined in such a way as to contain all objects that participate in any non-negligible [conservative|conservative force] interactions that are present. {note}For example, for systems subject to earth's gravity, the earth should technically be included in the system, though it is usually sufficient to treat it as a rigid body that is at rest and has infinite mass. If this approximation is made, the earth will have zero kinetic energy (it will not change its velocity, since it has infinite mass).{note} \\ *[Interactions|interaction]:*  All forces that do [non-conservative] [work] on the system must be considered, _including_ [internal forces|internal force] that perform such work. [Conservative forces|conservative force] that are present should have their interaction represented by a [potential energy] rather than by [work]. \\ h4. Descriptors *[Object Variables|object variable]*:  None. {note}Object masses and moment of inertia can technically change in this model, so they are state variables.{note} *[State Variables|state variable]*:  Mass (_m_^j^) and possibly moment of inertia (_I_{^}j{^}) for each object plus linear (_v_^j^) and possibly rotational (ω^j^) speeds for each object, or alternatively, the kinetic energy (_K_^j^) may be specified directly. If non-conservative forces are present, each object's vector position (_x_^j^) must be known *throughout* the time interval of interest (the path must be specified) unless the work done by each force is specified directly. When a conservative interaction is present, some sort of specific position or separation is required for each object (height _h_^j^ for near-earth [gravity], separation _r_~jk~ for universal gravity, separation _x_~jk~ for an elastic interaction, etc.) unless the potential energy (_U_^jk^) is specified directly. Alternately, in place of separate kinetic and potential energies, the mechanical energy of the system (_E_) can be specified directly. *[Interaction Variables|interaction variable]*:  Relevant non-conservative forces (_F_^NC,j^~k~) or the work done by the non-conservative forces (_W_^NC,j^~k~). ---- h2. Model Equations h4. Relationships Among State Variables {latex}\begin{large}\[ E = K^{\rm sys} + U^{\rm sys} \] \[ K^{\rm sys} = \sum_{j=1}^{N}\left(\frac{1}{2}m^{j}(v^{j})^

{2} + \frac{1}{2}

I^{j}(

I\omega^

{j})^

{2}\

right)\] \[ W^{NC,j}_{k

]\end{large}

Column

Work

Latex
\begin{large}\[W_{fi} = \int_{\rm path} \vec{F}

^

(\vec{

NC}_{k}

s}) \cdot d\vec{

x}^{j} \] \[ W^{NC,{\rm sys}}

s} = \

sum

int_{

j=1}^{N} \sum_{k=1

t_{i}}^{

N

t_{

F,j

f}}

W^{NC,j}_{k}

\vec{F}(t) \cdot \vec{v}(t)\:dt\]\end{large}

{latex} where _N_ is the number of system constitutents and _N_~F,j~ is the number of non-conservative forces acting on the _j{_}th system constitutent. The system potential energy is the sum of all the potential energies produced by interactions between system constituents. {warning}There are always two system constituents involved in any interaction. It is technically important to count each *interaction* only once, even though two constituents are involved.{note} Some common potential energy relationships are: h5. Near-Earth Gravity {latex}

Note
The system potential energy is the sum of all the potential energies produced by interactions between system constituents. Even when there are two system constituents involved (for example in a double star) each interaction produces only one potential energy.

S.I.M. Structure of the Model

Compatible Systems

One or more point particles or rigid bodies, plus any conservative interactitons that can be accounted for as potential energies of the system.

Info
In mechanics, the only commonly encountered conservative interactions are gravity and springs.

Relevant Interactions

Any external force that performs that perform work on the system must be considered, and also any internal non-conservative forces that perform work. Any internal conservative forces that are present should have their interaction represented by the associated potential energy rather than by the work.

Law of Change

Mathematical Representation

Section

Column

Differential Form

Latex
\begin{large}\[

U_

\frac{

g

dE}

^

j

dt} =

m^{j}gh^{j}

\sum \

]

left(\

end

vec{

large

F}^{

latex} h5. Universal Gravity {latex}\begin{large}\[ U_{g}^{jk} = -G\frac{m^{j}m^{k}}{|\vec{r}_{jk}}}

\rm ext} + \vec{F}^{\rm NC}\right)\cdot \vec{v} \]\end{large}

{latex} h5. Elastic {latex}

Column

Integral Form

Latex
\begin{large}\[

U

E_{

s}^{jk

f} =

\frac{1}{2}k^{jk}(\vec{x}^{j}-\vec{x}^{k}-x_{0}^{jk})^{2}\]\end{large}{latex} where _x_^jk^~0~ is the natural length of the spring. h4. Mathematical Statement of the Model {latex} \begin {large} $E_{f} = E_{i} + W^{NC,{\rm sys}} $ \end{large}{latex}\\ where _W{^}NC{^}{~}i,f{~}_ is the [work] done by the all the non-conservative forces on the system between the initial state defined by _E{~}i{~}_ and the final state defined by _E{~}f{~}_ and is given by {latex}\begin{large}$ W_{i,f}^{NC} = \int_{i}^{f} \sum \vec{F}^{NC} . d\vec{r} $ \end{large}{latex} | !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. | \\

E_{i} + \sum W^{\rm ext}_{fi} + \sum W^{\rm NC}_{fi} \] \end{large}

Diagrammatic Representations

Relevant Examples

Toggle Cloak
id cons
Examples Involving Constant Mechanical Energy

Cloak

id	cons

50falsetrueAND

constant_energy,example_problem

Toggle Cloak
id noncons
Examples Involving Non-Conservative Work

Cloak

id	noncons

50falsetrueAND

non-conservative_work,example_problem

Toggle Cloak
id grav
Examples Involving Gravitational Potential Energy

Cloak

id	grav

50falsetrueAND

gravitational_potential_energy,example_problem

Toggle Cloak
id elas
Examples Involving Elastic (Spring) Potential Energy

Cloak

id	elas

50falsetrueAND

elastic_potential_energy,example_problem

Toggle Cloak
id rot
Examples Involving Rotational Kinetic Energy

Cloak

id	rot

50falsetrueAND

rotational_energy,example_problem

Toggle Cloak
id all
All Examples Using this Model

Cloak

id	all

50falsetrueAND

constant_energy,example_problem

50falsetrueAND

non-conservative_work,example_problem

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Child pages

Versions Compared

Old Version 19

New Version Current

Key

Introduction to the Model

Description and Assumptions

Learning Objectives

Relevant Definitions

Mechanical Energy

Kinetic Energy

Work

S.I.M. Structure of the Model

Compatible Systems

Relevant Interactions

Law of Change

Mathematical Representation

Differential Form

Integral Form

Diagrammatic Representations

Relevant Examples

Toggle Cloak
id cons
Examples Involving Constant Mechanical Energy

Toggle Cloak
id noncons
Examples Involving Non-Conservative Work

Toggle Cloak
id grav
Examples Involving Gravitational Potential Energy

Toggle Cloak
id elas
Examples Involving Elastic (Spring) Potential Energy

Toggle Cloak
id rot
Examples Involving Rotational Kinetic Energy

Toggle Cloak
id all
All Examples Using this Model

Child pages

Page History

Versions Compared

Old Version 19

New Version Current

Key

Introduction to the Model

Description and Assumptions

Learning Objectives

Relevant Definitions

Mechanical Energy

Kinetic Energy

Work

S.I.M. Structure of the Model

Compatible Systems

Relevant Interactions

Law of Change

Mathematical Representation

Differential Form

Integral Form

Diagrammatic Representations

Relevant Examples

Toggle Cloakidcons Examples Involving Constant Mechanical Energy

Toggle Cloakidnoncons Examples Involving Non-Conservative Work

Toggle Cloakidgrav Examples Involving Gravitational Potential Energy

Toggle Cloakidelas Examples Involving Elastic (Spring) Potential Energy

Toggle Cloakidrot Examples Involving Rotational Kinetic Energy

Toggle Cloakidall All Examples Using this Model

Toggle Cloak
id cons
Examples Involving Constant Mechanical Energy

Toggle Cloak
id noncons
Examples Involving Non-Conservative Work

Toggle Cloak
id grav
Examples Involving Gravitational Potential Energy

Toggle Cloak
id elas
Examples Involving Elastic (Spring) Potential Energy

Toggle Cloak
id rot
Examples Involving Rotational Kinetic Energy

Toggle Cloak
id all
All Examples Using this Model