Question

Two particles of masses $m_1$ and $m_2$ ($m_1 > m_2$) are separated by a distance 'd'. When the positions of the two particles are interchanged, the shift in the centre of mass is:

• A. $\left(\frac{m_1 + m_2}{m_1 - m_2}\right)d$
• B. $\left(\frac{m_1 - m_2}{m_1 + m_2}\right)d$
• C. zero
• D. $\left(\frac{m_1}{m_1 - m_2}\right)d$

Hint:

The CM always shifts towards the heavier mass when they are swapped.

Answer 1

The Correct Option is B

Step 1: Set up positions.
Two masses $m_1$ and $m_2$ (with $m_1 > m_2$) sit a distance $d$ apart. Place $m_1$ at position $0$ and $m_2$ at position $d$ on a line.

Step 2: Recall the centre of mass formula.
The centre of mass position is the mass weighted average: $x_{cm} = \dfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$.

Step 3: Find the first centre of mass.
With $m_1$ at $0$ and $m_2$ at $d$: $x_{cm,1} = \dfrac{m_1(0) + m_2(d)}{m_1 + m_2} = \dfrac{m_2 d}{m_1 + m_2}$.

Step 4: Swap the masses.
Now put $m_2$ at $0$ and $m_1$ at $d$: $x_{cm,2} = \dfrac{m_2(0) + m_1(d)}{m_1 + m_2} = \dfrac{m_1 d}{m_1 + m_2}$.

Step 5: Find the shift.
The shift is the difference of the two centre positions: $|x_{cm,2} - x_{cm,1}| = \dfrac{(m_1 - m_2) d}{m_1 + m_2}$.

Step 6: Interpret it.
The centre of mass slides toward the side where the heavier mass now sits, by this amount. \[ \boxed{\left(\dfrac{m_1 - m_2}{m_1 + m_2}\right) d} \]