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Step-by-Step Solution
Step 1: Recall the Concept of Center of Mass
The position of the center of mass (COM) of a system of two particles with masses
$m_1$ and $m_2$ located at positions $x_1$ and $x_2$ respectively (along a single axis
for simplicity) is given by:
$$
x_{\mathrm{COM}} \;=\; \frac{m_1\,x_1 + m_2\,x_2}{m_1 + m_2}.
$$
Step 2: Initial Condition of the System
Assume initially the COM is at the origin, i.e., $x_{\mathrm{COM}} = 0$. Thus,
$$
0 \;=\; \frac{m_1\,x_1 + m_2\,x_2}{m_1 + m_2}
\quad\Longrightarrow\quad
m_1\,x_1 \;=\; -\,m_2\,x_2.
$$
In many representations, you might set one mass on the negative side ($-x_1$)
and the other on the positive side ($x_2$). However, the essential relationship is
$m_1\,x_1 = m_2\,x_2$ in magnitude when the COM is at the origin.
Step 3: Shift in the Position of $m_1$ and Required Shift for $m_2$
Suppose mass $m_1$ is moved towards the center by a distance $d$. That means
if $m_1$ was originally at $x_1$, its new position is $x_1 - d$. Let the distance
that $m_2$ must move (to keep the COM fixed at the origin) be $d'$. Hence
if $m_2$ was at $x_2$, its new position is $x_2 - d'$.
Step 4: Write the COM Condition After the Shift
After shifting, the COM should remain at $0$. Therefore:
$$
0 \;=\; \frac{m_1(x_1 - d) + m_2(x_2 - d')}{m_1 + m_2}.
$$
This implies
$$
m_1(x_1 - d) + m_2(x_2 - d') \;=\; 0.
$$
Step 5: Substitute the Known Relationship
From the initial condition, $m_1\,x_1 + m_2\,x_2 = 0 \;\Longrightarrow\; m_1\,x_1 = -\, m_2\,x_2.$
Using this in the above equation:
$$
m_1\,x_1 - m_1\,d + m_2\,x_2 - m_2\,d' = 0.
$$
Since $m_1\,x_1 + m_2\,x_2 = 0$, we can replace $m_1\,x_1 + m_2\,x_2$ by 0. So,
$$
-m_1\,d - m_2\,d' = 0
\quad\Longrightarrow\quad
m_2\,d' = m_1\,d.
$$
Step 6: Final Result for the Required Distance
Solving for $d'$, we get
$$
d' \;=\; \frac{m_1}{m_2}\,d.
$$
Thus, if the first mass $m_1$ is moved a distance $d$ towards the COM, the second mass
$m_2$ must be moved by the distance $\frac{m_1}{m_2}\,d$ in order to keep the
center of mass fixed at its original location.
