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
{
Wiki Markup
Composition Setup
HTML Table
border1
cellpadding8
cellspacing0
rulescols
framevoid
}{composition-setup} {table:border=1|frame=void|rules=cols|cellpadding=8|cellspacing=0} {tr:valign=top} {td:width=250|bgcolor=#F2F2F2} {live-template:Left Column} {td} {td} h1. Gravity {excerpt}The [gravitational force|gravitation (universal)] exerted by the earth on an object near the earth's surface.{excerpt} h3. The Force of Gravity Near Earth's Surface h4. Defining "Near" Suppose an object of mass {*}_m_{*} is at a height {*}_h_{*} above the surface of the earth. Assume that the earth is spherical with radius {*}_R{_}{~}E~{*}. Working in spherical coordinates with the origin at the center of the earth, the gravitational force on the object from the earth will be: \\ {latex}
Table Row (tr)
valigntop
Table Cell (td)

Excerpt

The gravitational force exerted by the earth on an object near the earth's surface.

The Force of Gravity Near Earth's Surface

Defining "Near"

Suppose an object of mass m is at a height h above the surface of the earth. Assume that the earth is spherical with radius RE. Working in spherical coordinates with the origin at the center of the earth, the gravitational force on the object from the earth will be:

Latex
\begin{large}\[ \vec{F} = - G \frac{M_{E}m}{(R_{E}+h)^{2}} \hat{r} \]\end{large}
{latex} \\ A Taylor expansion gives: \\ {latex}


A Taylor expansion gives:

Latex
\begin{large}\[ \vec{F} \approx - G \frac{M_{E}m}{R_{E}^{2}}\left(1 - 2\frac{h}{R_{E}} + ...\right)\hat{r} \]\\
\end{large}
{latex} \\


Thus,

for

{*}_

h

_

/

_R{_}{~}E~ << 1{*}, the gravitational force from the earth on the object will be essentially independent on altitude above the earth's surface and will have a magnitude equal to: \\ {latex}

RE << 1, the gravitational force from the earth on the object will be essentially independent on altitude above the earth's surface and will have a magnitude equal to:

Latex
\begin{large}\[ F_{g} = mG\frac{M_{E}}{R_{E}^{2}} \]\end{large}
{latex} h4. Defining {*}_g_{*} The above expression is of the form: \\ {latex}

Defining g

The above expression is of the form:

Latex
\begin{large}\[ F_{g} = mg \]\end{large}
{latex} \\ if we take: \\ {latex}


if we take:

Latex
\begin{large}\[ g = G\frac{M_{E}}{R_{E}^{2}} = \left(6.67\times 10^{-11}\mbox{ N}\frac{\mbox{m}^{2}}{\mbox{kg}^{2}}\right)\left(\frac{5.98\times 10^{24}\mbox{ kg}}{(6.37\times 10^{6}\mbox{ m})^{2}}\right) = \mbox{9.8 m/s}^{2}\]\end{large}
{latex} h3. Gravitational Potential Energy Near

Gravitational Potential Energy Near Earth's

Surface

Near

the

earth's

surface,

if

we

assume

coordinates

with

the

{*}

+

{_}

y

_{*}

direction

pointing

upward,

the

force

of

gravity

can

be

written:

{

Latex
}
\begin{large}\[ \vec{F} = -mg \hat{y}\]\end{large}
{latex}

Since

the

"natural"

ground

level

varies

depending

upon

the

specific

situation,

it

is

customary

to

specify

the

coordinate

system

such

that:

{

Latex
}
\begin{large}\[ U(0) \equiv 0\]\end{large}
{latex}

The

gravitational

potential

energy

at

any

other

height

{*}_

y

_{*}

can

then

be

found

by

choosing

a

path

for

the

work

integral

that

is

perfectly

vertical,

such

that:

{

Latex
}
\begin{large}\[ U(y) = U(0) - \int_{0}^{y} (-mg)\;dy = mgy\]\end{large}
{latex}

For

an

object

in

vertical

freefall

(no

horizontal

motion)

the

associated

[

potential

energy

]

curve

would

then

be:

!nearearth.gif! For movement under pure

Image Added

For movement under pure near-earth

gravity,

then,

there

is

no

equilibrium

point.

At

least

one

other

force,

such

as

a

normal

force,

tension,

etc.,

must

be

present

to

produce

equilibrium.

h3. Example Problems involving

Example Problems involving Near-earth

Gravity h4. {

Gravity

Toggle Cloak

:

id

=

force

}

Example

Problems

involving

Gravitational

Force

{

Cloak

:
id
=
force
} {contentbylabel:

Content by Label
gravity,example_problem
gravity,example_problem
|
maxResults
=50|operator=AND|showSpace=false|excerpt=true} {cloak:force} h4. {toggle-cloak:id=en} Example Problems involving Gravitational Potential Energy {cloak:id=en} {contentbylabel:
50
showSpacefalse
excerpttrue
operatorAND
excerptTypesimple
cqllabel = "example_problem" and label = "gravity"
Cloak
force
force

Toggle Cloak
iden
Example Problems involving Gravitational Potential Energy

Cloak
iden

Content by Label
gravitational_potential_energy,example_problem
gravitational_potential_energy,example_problem
|
maxResults
=50|operator=AND|showSpace=false|excerpt=true} {cloak:en} {td} {tr} {table} {live-template:RELATE license}
50
showSpacefalse
excerpttrue
operatorAND
excerptTypesimple
cqllabel = "gravitational_potential_energy" and label = "example_problem"
Cloak
en
en