Question

The center of mass of three particles of masses 1kg, 2kg and 3kg is located at the point $(3,3,3)$. Where should the fourth particle of mass 4kg be positioned so that the center of mass shifts to $(1,1,1)$.

Answer 1

The center of mass of the initial three particle system is known to us. Also the final center of mass is known to us. So, the final center of mass should be the combination of center of mass of the three-particle system and the fourth particle whose position is to be found.

Complete answer:
Since, the combined center of mass of all the three particles is given. Let that point be denoted by M. Then,
$\Rightarrow M\equiv (3,3,3)$
We will assume that the combined mass of the three particles that is 1kg, 2kg and 3kg is at this point M.
Now let the new center of mass be denoted by N. Then it has been given to us that:
$\Rightarrow N\equiv (1,1,1)$
Let the position of the fourth particle of 4kg be denoted by O. And let its coordinates be given by:
$\Rightarrow O\equiv (x,y,z)$
Now using the formula of Center of mass for a given number of particles, we have:
$\begin{aligned} & \Rightarrow 1=\dfrac{6\times (3)+4\times (x)}{10} \\\ & \Rightarrow 10=18+4x \\\ & \Rightarrow 4x=-8 \\\ & \Rightarrow x=-2 \\\ \end{aligned}$
Hence, the x-coordinate of the fourth particle comes out to be (-2).
Similarly, for the y-coordinate we have:
$\begin{aligned} & \Rightarrow 1=\dfrac{6\times (3)+4\times (y)}{10} \\\ & \Rightarrow 10=18+4y \\\ & \Rightarrow 4y=-8 \\\ & \Rightarrow y=-2 \\\ \end{aligned}$
Hence, the y-coordinate of the fourth particle comes out to be (-2).
Similarly, for the z-coordinate we have:
$\begin{aligned} & \Rightarrow 1=\dfrac{6\times (3)+4\times (z)}{10} \\\ & \Rightarrow 10=18+4z \\\ & \Rightarrow 4z=-8 \\\ & \Rightarrow z=-2 \\\ \end{aligned}$
Hence, the z-coordinate of the fourth particle comes out to be (-2).
Hence, the position where the fourth particle should be placed to shift the Center of Mass of the whole system from $(3,3,3)$ to $(1,1,1)$ is $(-2,-2,-2)$ .

Note:
In this problem, we went out and calculated all the three coordinates of the fourth particle separately and found out that all of their magnitude was the same. Another way to analyze this fact was by seeing the fact that all the three coordinates of the system were equal initially and afterwards, so the fourth particle must be placed on the line joining them, that is, all of its three coordinates must be the same too.