Versions Compared

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin

Excerpt

The relationship between the kinetic energy of a point particle and the work done on the point particle. This theorem is one way to arrive at a mathematical definition of work.

Statement of the Theorem

If all the influences on a point particle are represented as works, the net work done by the forces produces a change in the kinetic energy of the particle according to:

Latex
Wiki Markup
{excerpt}The relationship between the [kinetic energy] of a [point particle] and the [work] done on the point particle.  This theorem is one way to arrive at a mathematical definition of work.{excerpt}

h4. Statement of the Theorem

If all the influences on a [point particle] are represented as [works|work], the net work done by the forces produces a change in the kinetic energy of the particle according to:

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

h4. Derivation of the Theorem

From [

Derivation of the Theorem

From Newton's

...

2nd

...

Law

...

for

...

a

...

point

...

particle,

...

we

...

know

{
Latex
}\begin{large}\[ \vec{F}_{\rm net} = m\frac{d\vec{v}}{dt}\]\end{large}{latex}

Now

...

suppose

...

that

...

the

...

particle

...

undergoes

...

an

...

infinitesimal

...

displacement

...

dr

...

.

...

Since

...

we

...

want

...

to

...

bring

...

the

...

left

...

side

...

of

...

the

...

equation

...

into

...

line

...

with

...

the

...

form

...

of

...

the

...

expression

...

for

...

work,

...

we

...

take

...

the

...

dot

...

product

...

of

...

each

...

side

...

with

...

the

...

displacement:

{
Latex
}\begin{large}\[ \vec{F}_{\rm net}\cdot d\vec{r} = m\frac{d\vec{v}}{dt} \cdot d\vec{r} \]\end{large}{latex}

Before

...

we

...

can

...

integrate,

...

we

...

make

...

a

...

substitution.

...

Since

...

v

...

is

...

the

...

velocity

...

of

...

the

...

particle,

...

we

...

can

...

re-express

...

the

...

infinitesimal

...

displacement

...

as:

{
Latex
}\begin{large}\[ d\vec{r} = \vec{v}dt\]\end{large}{latex}

Making

...

this

...

substitution

...

on

...

the

...

right

...

hand

...

side

...

of

...

the

...

equation,

...

we

...

have:

{
Latex
}\begin{large}\[ \vec{F}_{\rm net}\cdot d\vec{r} = m\frac{d\vec{v}}{dt}\cdot \vec{v}\:dt = m\vec{v}\cdot d\vec{v} = m(v_{x}\;dv_{x} + v_{y}\;dv_{y}+v_{z}\;dv_{z})\]\end{large}{latex}

We

...

can

...

now

...

integrate

...

over

...

the

...

path:

{
Latex
}\begin{large}\[ \int_{\rm path} \vec{F}_{\rm net} \cdot d\vec{r} = \frac{1}{2}m(v_{x,f}^{2}-v_{x,i}^{2} + v_{y,f}^{2} - v_{y,i}^{2} + v_{z,f}^{2} - v_{z,i}^{2}) = \frac{1}{2}m(v_{f}^{2}-v_{i}^{2})\]\end{large}{latex}

which

...

is

...

equivalent

...

to

...

the

...

Work-Kinetic

...

Energy

...

Theorem.

...