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Mechanical

...

Energy

...

Excerpt

...

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

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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:

Latex



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

...

introductory

...

mechanics,

...

it

...

is

...

basically

...

assumed

...

that

...

the

...

possible

...

constituents

...

of

...

the

...

potential

...

energy

...

are

...

gravitational

...

potential

...

energy

...

(

...

U

...

g)and

...

elastic

...

potential

...

energy

...

(

...

U

...

e),

...

so

...

that

...

the

...

mechanical

...

energy

...

is

...

essentially:

{
Latex
}\begin{large}\[ E = K + U_{g} + U_{e}\]\end{large}{latex}

h3. Generalized 

Generalized Work-Energy

...

Theorem

...

The

...

Work-Kinetic

...

Energy

...

Theorem

...

states:

{
Latex
}\begin{large}\[ \Delta K = W_{\rm net}\]\end{large}{latex}

where _W_~net~ 

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
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

...

that

...

the

...

right

...

hand

...

side

...

is

...

the

...

sum

...

of

...

the

...

works

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arising

...

from

...

all

...

forces

...

that

...

do

...

not

...

have

...

an

...

associated

...

potential

...

energy.

...

Conditions

...

for

...

Mechanical

...

Energy

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Conservation

...

From

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the

...

generalized

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Work-Energy

...

Theorem,

...

we

...

see

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that

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the

...

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

...

mechanical

...

interactions)

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when

...

the

...

net

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non-conservative

...

work

...

done

...

on

...

the

...

system

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is

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zero.

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Since

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gravitation

...

(universal)

...

and

...

spring

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forces

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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

...

treated

...

in

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introductory

...

mechanics,

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this

...

condition

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usually

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amounts

...

to

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the

...

constraint

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that

...

the

...

total

...

work

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done

...

by

...

forces

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other

...

than

...

gravity

...

or

...

spring

...

forces

...

is

...

zero.

Note

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

_ [

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

|scalar product]

of

the

[

normal

force

]

with

the

path

will

always

be

zero

and

the

[

normal

force

]

will

contribute

zero

[

work

]

.

{note}