Consider a two particle system with particles having masses `m_1 and m_2` if the first particle is pushed towards the centre of mass through a distance d, by what distance should the second particle is moved, so as to keep the center of mass at the same positon?

Options

A. `(m_2)/(m_1) d`
B. `(m_1)/(m_1+m_2)d`
C. `(m_1)/(m_2)d`
D. d

Explanation

To solve the problem of how far the second particle should be moved to keep the center of mass in the same position after moving the first particle a distance \( d \) towards the center of mass, we can follow these steps:

### Step 1: Define the initial positions and center of mass
Let the position of the first particle (mass \( m_1 \)) be at the origin, i.e., \( x_1 = 0 \). The position of the second particle (mass \( m_2 \)) is at \( x_2 = l \). The center of mass \( x_{cm} \) of the system is given by the formula:

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

Substituting the values, we have:

\[ x_{cm} = \frac{m_1 \cdot 0 + m_2 \cdot l}{m_1 + m_2} = \frac{m_2 l}{m_1 + m_2} \]

### Step 2: Move the first particle
Now, the first particle is moved towards the center of mass by a distance \( d \). The new position of the first particle becomes:

\[ x_1' = 0 + d = d \]

### Step 3: Define the new position of the second particle
Let the second particle be moved by a distance \( x \) towards the first particle. Therefore, its new position will be:

\[ x_2' = l - x \]

### Step 4: Calculate the new center of mass
Now, we need to find the new center of mass \( x_{cm}' \) after moving both particles:

\[ x_{cm}' = \frac{m_1 x_1' + m_2 x_2'}{m_1 + m_2} \]

Substituting the new positions, we have:

\[ x_{cm}' = \frac{m_1 d + m_2 (l - x)}{m_1 + m_2} \]

### Step 5: Set the new center of mass equal to the original center of mass
Since we want the center of mass to remain in the same position, we set \( x_{cm}' = x_{cm} \):

\[ \frac{m_1 d + m_2 (l - x)}{m_1 + m_2} = \frac{m_2 l}{m_1 + m_2} \]

### Step 6: Simplify the equation
Multiplying both sides by \( m_1 + m_2 \) gives:

\[ m_1 d + m_2 (l - x) = m_2 l \]

Expanding and rearranging terms results in:

\[ m_1 d + m_2 l - m_2 x = m_2 l \]

This simplifies to:

\[ m_1 d = m_2 x \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):

\[ x = \frac{m_1 d}{m_2} \]

### Conclusion
Thus, the second particle should be moved a distance of \( \frac{m_1 d}{m_2} \) towards the first particle to keep the center of mass at the same position.

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