Say you have 2 mass particles m1 and m2 about some cartesian coordinate system whose origin is at position A , while another at position B .
How would one prove that the position of COM of the particle system is independent of origin A or B?
2
-
$\begingroup$ Consider the distances between m1, m2, and the COM, in both coordinate frames. $\endgroup$– PM 2RingCommented Aug 23 at 12:46
-
$\begingroup$ See PM 2Ring comment. $\endgroup$– Bob DCommented Aug 23 at 13:14
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
Let's assume that the co-ordinates of the A-frame and B-frame are related by a translation as follows:
$$\vec{r_A} = \vec{r_B} +\vec{a}$$
Then we have to prove that the coordinates of the CoM with respect to the A and the B frames are also related in the same manner.
By definition, in frame A we have
$$\vec{R_A} = \frac{m_1 \vec{r_{1A}}+m_2 \vec{r_{2A}}}{m_1+m_2} = \frac{m_1 \vec{r_{1B}}+ m_2 \vec{r_{2B}}}{m_1+m_2} +\vec{a} = \vec{R_B} +\vec{a}$$
where in the intermediate step we have used the first equation.
-
$\begingroup$ Many thanks and you made it really easy for me to understand. $\endgroup$– DubiousCommented Aug 23 at 21:49