# Centre of mass

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:

Doc Al
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.)

Doc Al said:
$$\sum_{j \neq i}F_i_j = \sum_{i < j}(F_j_i + F_i_j)$$

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

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

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

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

is coming from.

Doc Al
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.