Versions Compared

Old Version 2

changes.mady.by.user David E Pritchard

Saved on

compared with

New Version Current

changes.mady.by.user Andrew E Pawl

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.
Comment: Migration of unmigrated content due to installation of a new plugin
Composition Setup
 Excerpt
The average position of the mass in a body or system.  A system will behave in response to external forces applied to any of its parts as if the entire mass of the body were concentrated there.  The motion of the center of mass is unaffected by internal forces in the system (e.g. forces between the atoms, or collisions between different components of the system).

Motivation for Concept
While the motion of a point particle in response to well-defined external forces is relatively straightforward to determine, it is not so simple to calculate the motion of an extended body, especially if it is not rigid. The motion of the center of mass of a body, however, will act as if all of the mass of the body were concentrated there as a point particle, and will act the same way a point particle will in response to the same external forces. This is true even if the body is not rigid, if it is rotating, and regardless of where the forces are applied (e.g. even at the edge of the body, or to only one part of the system). This is therefore a very useful concept fot visualizing and calculating the dynamics of bodies that are not themselves point masses.
Mathematical Definition
Body as Sum of Point Particles
Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the body to be a collection of Np point particles. Each of the Np point particles (of mass mi where i runs from 1 to Np) will obey [Newton's 2nd Law|Newton's Second Law
 Cloak
body
body

Center of Mass as a Sum
We define the Center of Mass as:
 Latex
 Wiki Markup
{composition-setup}{composition-setup}
{table:border=1|frame=void|rules=cols|cellspacing=0|cellpadding=8}
{tr:valign=top}
{td:width=275px|bgcolor=#F2F2F2}
{live-template:Left Column}
{td}
{td}

h1. Center of Mass
{excerpt}The average position of the mass in a body or system.  A system will behave in response to external forces applied to any of its parts as if the entire mass of the body were concentrated there.  The motion of the center of mass is unaffected by internal forces in the system (e.g. forces between the atoms, or collisions between different components of the system).
{excerpt}
\\

h3. {toggle-cloak:id=mot} Motivation for Concept

{cloak:id=mot}
While the motion of a [point particle] in response to well-defined external forces is relatively straightforward to determine, it is not so simple to calculate the motion of an extended body, especially if it is not rigid. The motion of the *center of mass* of a body, however, will act as if all of the mass of the body were concentrated there as a point particle, and will act the same way a point particle will in response to the same external forces. This is true even if the body is not rigid, if it is rotating, and regardless of where the forces are applied (e.g. even at the edge of the body, or to only one part of the system). This is therefore a very useful concept fot visualizing and calculating the dynamics of bodies that are not themselves point masses.
{cloak}

h3. {toggle-cloak:id=def}Mathematical Definition

{cloak:id=def}
{anchor:der}

h4. {toggle-cloak:id=body} Body as Sum of Point Particles

{cloak:id=body}
Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the body to be a collection of _N{_}{~}p~ [point particles|point particle]. Each of the _N{_}{~}p~ point particles (of mass _m{_}{~}i~ where i runs from 1 to _N{_}{~}p~) will obey \[Newton's 2nd Law\|Newton's Second Law
{cloak:body}\\

h4. {toggle-cloak:id=momsum} Center of Mass as a Sum

{cloak:id=momsum}
{anchor:sum}
We define the Center of Mass as:
{latex}\begin{large} \[ \vec{CoM} = \frac{\sum_{i=1}^{N_{\rm p}} m_{i}\vec{r_{i}}}{\sum_{i=1}^{N_{\rm p}} m_{i}} \] \end{large}{latex}
{cloak:momsum}
{cloak:def}

h3. {toggle-cloak:id=calc}Calculating Center of Mass

{cloak:id=calc}

h4. {toggle-cloak:id=intrec}Integrals in Rectangular Coordinates

{cloak:id=intrec}
For _continuous_ objects, the summation in our [definition|#sum] of the center of mass must be converted to an integral.  The Mass differential is given by:
{latex}
Calculating Center of Mass
Integrals in Rectangular Coordinates
For continuous objects, the summation in our definition of the center of mass must be converted to an integral.  The Mass differential is given by:
 Latex
\begin{large}\[ dm = \rho(x,y,z) dx dy dz \] \end{large}{latex}

To 
...
 calculate 
...
 the 
...
x
...
 - 
...
 position 
...
 of 
...
 the 
...
 center 
...
 of 
...
 mass 
...
 we 
...
 use:

{
 Latex
}\begin{large}\[ CoM_{\rm x} = \frac{\int \int \int \:\:x\:\rho(x,y,z)\:dx\:dy\:dz}{\int \int \int \:\rho(x,y,z)\:dx\:dy\:dz} \]\end{large}{latex}

We 
...
 can 
...
 obtain 
...
 the 
...
y
...
 - 
...
 and 
...
z
...
 - 
...
 coordinates 
...
 using 
...
 similar 
...
 expressions:

{
 Latex
}\begin{large}\[ CoM_{\rm y} = \frac{\int \int \int \:\:y\:\rho(x,y,z)\:dx\:dy\:dz}{\int \int \int \:\rho(x,y,z)\:dx\:dy\:dz} \]\end{large}

 Latex
{latex}
\\
{latex}\begin{large}\[ CoM_{\rm z} = \frac{\int \int \int \:\:z\:\rho(x,y,z)\:dx\:dy\:dz}{\int \int \int \:\rho(x,y,z)\:dx\:dy\:dz} \]\end{large}{latex}
{cloak:intrec}
{cloak:calc}
{td}
{tr}
{table}
\\
{live-template:RELATE license}