Centre of mass

1. Dec 29, 2005

ElDavidas

I was looking over my notes for centre of mass for a system and it says:

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

where M is the total mass of the system.

Then it considers the centre of Mass in motion:

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

$$= \sum_{i}E_i + \sum_{j \neq i}F_i_j = E + \sum_{i < j}(F_j_i + F_j_j)$$

$$= E$$

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

$$E + \sum_{i < j}(F_j_i + F_j_j)$$

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

Any help would be grateful!

Last edited: Dec 29, 2005
2. Dec 29, 2005

Staff: Mentor

Could there be a typo? I'd think that:
$$\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)$$

(And this term disappears due to Newton's 3rd law.)

3. Jan 1, 2006

ElDavidas

Is this something you just need to know? It suddenly appears in my Uni notes without any explanation.

4. Jan 1, 2006

Staff: Mentor

I'm not sure what you're asking. Are you asking "Why is that true?" or "How am I supposed to know that?"

5. Jan 1, 2006

ElDavidas

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

$$\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)$$

is coming from.

6. Jan 1, 2006

Staff: Mentor

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.