Centre of mass

  • Thread starter ElDavidas
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  • #1
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I was looking over my notes for centre of mass for a system and it says:

[tex] c = \frac {1} {M} \sum_{i} m_i\ddot{r}_i = \sum_{i}(E_i + \sum_{j \neq i}F_i_j) [/tex]

where M is the total mass of the system.

Then it considers the centre of Mass in motion:

[tex] M \ddot{c} = \sum_i m_i \ddot{r}_i = \sum_{i}(E_i + \sum_{j \neq i}F_i_j) [/tex]

[tex] = \sum_{i}E_i + \sum_{j \neq i}F_i_j = E + \sum_{i < j}(F_j_i + F_j_j)[/tex]

[tex] = E [/tex]

The thing is, I don't understand the line:

[tex]E + \sum_{i < j}(F_j_i + F_j_j)[/tex]

and how it comes about (especially the i < j) part.

Any help would be grateful!
 
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Answers and Replies

  • #2
Doc Al
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Could there be a typo? I'd think that:
[tex]\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)[/tex]

(And this term disappears due to Newton's 3rd law.)
 
  • #3
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Doc Al said:
[tex]\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)[/tex]

Is this something you just need to know? It suddenly appears in my Uni notes without any explanation.
 
  • #4
Doc Al
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I'm not sure what you're asking. Are you asking "Why is that true?" or "How am I supposed to know that?"
 
  • #5
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Doc Al said:
I'm not sure what you're asking. Are you asking "Why is that true?" or "How am I supposed to know that?"

What I mean is why is this true? Can this be shown from previous statements regarding the interaction of forces. I don't understand where

[tex]\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)[/tex]

is coming from.
 
  • #6
Doc Al
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The truth of this has nothing to do with physics; it's just a mathematical truism. To see that it's true, make up a small example (where i,j < 5, say) and confirm that both sides are equal.
 

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