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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 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:

{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}\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:

\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:

_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}\begin{large}\[ F_{g} = mG\frac{M_{E}}{R_{E}^{2}} \]\end{large}{latex}

h4. Defining {*}_g_{*}

The above expression is of the form:
\\

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:

\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

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Example

Problems

involving

Gravitational

Force

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Example Problems involving Gravitational Potential Energy