AIPMT PRELIMS 2004 Physics Motion of Center of Mass MCQ Question

To solve the problem, we first need to understand the concept of the center of mass (COM) for a system of two particles. The center of mass of a two-particle system with masses $m_1$ and $m_2$ is given by the formula:

$$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$$

where $x_1$ and $x_2$ are the positions of the particles with respect to an arbitrary origin.

Let’s denote the initial positions of the two particles as $x_1$ (for mass $m_1$) and $x_2$ (for mass $m_2$). When we push mass $m_1$ towards the center of mass by a distance $d$, its new position becomes $x_1' = x_1 - d$.

To keep the center of mass at the same position, the new position of mass $m_2$ must change accordingly. Let’s denote the distance that mass $m_2$ moves as $x$. Therefore, the new position of mass $m_2$ will be $x_2' = x_2 + x$.

The position of the center of mass before the displacement is:

$$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$$

After displacing $m_1$ by $d$ and moving $m_2$ by $x$, the new center of mass $x_{cm}'$ is given by:

$$x_{cm}' = \frac{m_1 (x_1 - d) + m_2 (x_2 + x)}{m_1 + m_2}$$

Since we want the center of mass to remain unchanged, we set $x_{cm}' = x_{cm}$:

$$\frac{m_1 (x_1 - d) + m_2 (x_2 + x)}{m_1 + m_2} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$$

Multiplying through by $m_1 + m_2$ to eliminate the denominator:

$$m_1 (x_1 - d) + m_2 (x_2 + x) = m_1 x_1 + m_2 x_2$$

Expanding and rearranging terms gives:

$$m_1 x_1 - m_1 d + m_2 x_2 + m_2 x = m_1 x_1 + m_2 x_2$$

Simplifying this results in:

$$-m_1 d + m_2 x = 0$$

From this, we can express $x$ (the distance that mass $m_2$ moves) in terms of $d$:

$$m_2 x = m_1 d$$

Thus, we find:

$$x = \frac{m_1}{m_2} d$$

However, since we are looking for the distance $m_2$ moves in terms of $d$ and not just any arbitrary value, we rearrange this to:

$$x = \frac{m_1}{m_2} d$$

This means that the distance mass $m_2$ moves to keep the center of mass in the original position is $\frac{m_1}{m_2} d$, confirming that the correct answer is Option A: $\frac{m_1}{m_2} d$.

Why Other Options Are Incorrect:

• Option B: $d$ - This suggests that $m_2$ moves the same distance as $m_1$, which would not keep the center of mass fixed unless $m_1 = m_2$.
• Option C: $\frac{m_2}{m_1}$ - This does not relate to the distance moved and does not consider the displacement $d$.
• Option D: $\frac{m_1}{m_1 + m_2} d$ - This incorrectly applies the mass ratio to the displacement without considering the relative motion necessary to keep the center of mass in place.

In summary, the movement of the second mass $m_2$ has to account for the ratios of the masses to ensure the center of mass remains unchanged, leading us to the conclusion that the correct