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{excerpt}An interaction between two massive particles resulting in an attractive force exerted on each by the other.  The force is proportional to the gravitational constant {*}G = {{6.674 28(67) x 10{*}{*}^\-11{^}* *m{*}{*}{^}3{^}{*}* kg{*}{*}^\-1{^}{*}* s{*}{*}^\-2{^}{*}}}, and the masses of the bodies, and inversely proportional to the square of the distance between them.{excerpt}

h3. Motivation for Concept

Newton's Law of Universal Gravitation provides an effective description of the movement of objects from submillimeter distances to galactic sizes, and is the dominant force on most (macroscopic) astronomical bodies.

h3. Gravitational Force (Newton's Law of Universal Gravitation)


h4. Statement of the Law for Point Masses

Between any two point masses (masses {*}_m{_}{~}1~{*} and {*}_m{_}{~}2~{*}, respectively) there will exist an attractive force along the line joining the masses.  The force on body *1* due to body *2* will have the form:
{latex}
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 Excerpt
An interaction between two massive particles resulting in an attractive force exerted on each by the other.  The force is proportional to the gravitational constant G = 6.674 28(67) x 10-11 m3 kg-1 s-2, and the masses of the bodies, and inversely proportional to the square of the distance between them.
Motivation for Concept
Newton's Law of Universal Gravitation provides an effective description of the movement of objects from submillimeter distances to galactic sizes, and is the dominant force on most (macroscopic) astronomical bodies.
Gravitational Force (Newton's Law of Universal Gravitation)
Statement of the Law for Point Masses
Between any two point masses (masses m1 and m2, respectively) there will exist an attractive force along the line joining the masses. The force on body 1 due to body 2 will have the form:
 Latex
\begin{large}\[ \vec{F}_{12} = - G \frac{m_{1}m_{2}}{r_{12}^{2}} \hat{r}_{12} \]\end{large}
{latex}


where, 
 {*}_
G
_{*} 
 is 
 
 the 
 
 gravitational 
 
 constant 
 
 equal 
 
 to:

{
 Latex
}
\begin{large}\[ G = \mbox{6.67}\times\mbox{10}^{-11}\mbox{ N}\frac{\mbox{m}^{2}}{\mbox{kg}^{2}} \]\end{large}
{latex}

{*}_r{_}{~}12~{*} is the distance between the two objects and is obtained as the magnitude of the vector difference between the position vector of object {*}1{*} and the position vector of object {*}2{*}. The vector difference is expressed as:
\\
{latex}
r12 is the distance between the two objects and is obtained as the magnitude of the vector difference between the position vector of object 1 and the position vector of object 2. The vector difference is expressed as: 
 Latex
\begin{large}\[\vec{r}_{12} = \vec{r}_{1}-\vec{r}_{2}\]\end{large}
{latex}


h4. Compatibility with 
Compatibility with Newton's 
 
 Laws 
 
 of 
 
 Motion



Note 
 
 that 
 
 the 
 
 Universal 
 
 Law 
 
 of 
 
 Gravition 
 
 is 
 
 consistent 
 
 with 
 
 Newton's 
 [
Third 
 
 Law 
 
 of 
 
 Motion
|Newton's Third Law]:
\\
{latex}
: 
 Latex
\begin{large}\[ \vec{F}_{21} = -G\frac{m_{2}m_{1}}{r_{21}^{2}} \hat{r}_{21}\]\end{large}
{latex}
\\
Noting that the differences of the position vectors {*}_r{_}{~}12~{*} and {*}_r{_}{~}21~{*} will certainly satisfy:
\\
{latex}

 Noting that the differences of the position vectors r12 and r21 will certainly satisfy: 
 Latex
\begin{large}\[ \vec{r}_{12} = - \vec{r}_{21}\]\end{large}
{latex}
\\
which implies:
\\
{latex}

 which implies: 
 Latex
\begin{large}\[ \vec{F}_{12} = - \vec{F}_{21}\]\end{large}
{latex}

h4. The Case of Spherical Symmetry

Although the form of the Law of Universal Gravitation is strictly valid only for [point particles|point particle], it is possible to show that for extended objects with a spherically symmetric mass distribution, the Law will hold in the form stated above *provided that the positions of the spherical objects are specified by their centers*.

h3. Gravitational Potential Energy

h4. Form of the Potential Energy

For two spherically symmetric objects (objects {*}1{*} and {*}2{*}), it is customary to analyze the energy of the gravitational interaction by constructing spherical coordinates with one of the objects at the origin (if one of the objects dominates the mass of the system, its position is typically used as the origin).  Newton's Law of Universal Gravitation then takes the form:

{latex}
The Case of Spherical Symmetry
Although the form of the Law of Universal Gravitation is strictly valid only for point particles, it is possible to show that for extended objects with a spherically symmetric mass distribution, the Law will hold in the form stated above provided that the positions of the spherical objects are specified by their centers.
Gravitational Potential Energy
Form of the Potential Energy
For two spherically symmetric objects (objects 1 and 2), it is customary to analyze the energy of the gravitational interaction by constructing spherical coordinates with one of the objects at the origin (if one of the objects dominates the mass of the system, its position is typically used as the origin). Newton's Law of Universal Gravitation then takes the form:
 Latex
\begin{large}\[ \vec{F} = - G\frac{m_{1}m_{2}}{r^{2}} \hat{r}\]\end{large}
{latex}

where {*}_r_{*} is the position of the object that is not placed at the origin.

It is also customary to make the assignment that the potential energy of the system goes to zero as the separation goes to infinty:

{latex}
where r is the position of the object that is not placed at the origin.
It is also customary to make the assignment that the potential energy of the system goes to zero as the separation goes to infinty:
 Latex
\begin{large}\[ \lim_{r \rightarrow \infty} U(r) = 0 \]\end{large}
{latex}


Thus, 
 
 we 
 
 can 
 
 define 
 
 the 
 
 potential 
 
 for 
 
 any 
 
 separation 
 {*}_
r
_{*} 
 as:


{
 Latex
}
\begin{large}\[ U(r) = U(\infty) - \lim_{r_{0}\rightarrow \infty}\int_{r_{0}}^{r} \left(-G\frac{m_{1}m_{2}}{r^{2}}\right) \;dr 
= - Gm_{1}m_{2} \left(\frac{1}{r}-\lim_{r_{0}\rightarrow \infty}\frac{1}{r_{0}}\right)\]
\[U(r) = -G\frac{m_{1}m_{2}}{r} \]\end{large}
{latex}

h4. Potential Energy Curve

If the two objects are isolated from other influences, their [potential energy] curve is then:

!uvsr.gif!

This potential energy curve is somewhat misleading, since the potential is spherically symmetric.  Thus, although in spherical coordinates, {*}_r_{*} cannot go negative, if we define a one-dimensional coordinate system by following a radial line through the origin (suppose, for instance, we chose to follow the {*}_z_{*} axis where {*}_z_ = _r{_} cosθ{*}) we would generate a curve:

!uvsz.gif!

which indicates the possibility of stable equilibrium when the objects' separation goes to zero.  Of course, this is technically impossible for objects of finite size.

h4. Gravitational Potential Energy of a System

In a system composed of many spherically symmetric objects, the total gravitational potential energy can be found by adding up the contribution from each distinct _interaction_.  

It is very important to note that any pair of the bodies experiences only one interaction between them.  Take, for example, a system composed of four objects labeled {*}1{*}, {*}2{*}, {*}3{*} and {*}4{*}.  There are six distinct interactions among these bodies, each of which has an associated potential energy:

{latex}
Potential Energy Curve
If the two objects are isolated from other influences, their potential energy curve is then:
Image Added
This potential energy curve is somewhat misleading, since the potential is spherically symmetric. Thus, although in spherical coordinates, r cannot go negative, if we define a one-dimensional coordinate system by following a radial line through the origin (suppose, for instance, we chose to follow the z axis where z = r cosθ) we would generate a curve:
Image Added
which indicates the possibility of stable equilibrium when the objects' separation goes to zero. Of course, this is technically impossible for objects of finite size.
Gravitational Potential Energy of a System
In a system composed of many spherically symmetric objects, the total gravitational potential energy can be found by adding up the contribution from each distinct interaction. 
It is very important to note that any pair of the bodies experiences only one interaction between them. Take, for example, a system composed of four objects labeled 1, 2, 3 and 4. There are six distinct interactions among these bodies, each of which has an associated potential energy:
 Latex
\begin{large}\[ 1 \leftrightarrow 2 \mbox{ gives rise to }U_{12}\]
\[ 1 \leftrightarrow 3 \mbox{ gives rise to }U_{13}\]
\[ 1 \leftrightarrow 4 \mbox{ gives rise to }U_{14}\]
\[ 2 \leftrightarrow 3 \mbox{ gives rise to }U_{23}\]
\[ 2 \leftrightarrow 4 \mbox{ gives rise to }U_{24}\]
\[ 3 \leftrightarrow 4 \mbox{ gives rise to }U_{34}\]\end{large}
{latex}


The 
 
 total 
 
 potential 
 
 energy 
 
 would 
 
 then 
 
 be 
 
 given 
 
 by:


{
 Latex
}
\begin{large}\[ U_{\rm sys} = U_{12}+U_{13}+U_{14}+U_{23}+U_{24}+U_{34}\]\end{large}
{latex}

{warning}It is important to beware of the temptation to 
 Warning
It is important to beware of the temptation to double-count. 
  
 The 
 
 potential 
 
 energy 
 {*}_U_~12~{*} is associated with the interaction _between_ objects {*}1{*} and {*}2{*}, it is not associated separately with object {*}1{*} and with object {*}2{*}.{warning}

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 U12 is associated with the interaction between objects 1 and 2, it is not associated separately with object 1 and with object 2.
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