The centre of mass of two masses m & m' moves by distance `(x)/(5)` when mass m is moved by distance x and m' is kept fixed. The ratio `(m')/(m)` is

Options

A. 2
B. 4
C. `1//4`
D. None of these

Explanation

To solve the problem of finding the ratio \( \frac{m'}{m} \) given that the center of mass of two masses \( m \) and \( m' \) moves by a distance \( \frac{x}{5} \) when mass \( m \) is moved by a distance \( x \) and mass \( m' \) is kept fixed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Center of Mass Formula:**
The center of mass \( x_{cm} \) for two masses \( m \) and \( m' \) located at positions \( x_1 \) and \( x_2 \) respectively is given by:
\[ x_{cm} = \frac{m \cdot x_1 + m' \cdot x_2}{m + m'} \]

2. **Setting Up the Problem:**
Let's assume:
- Mass \( m \) is initially at position \( x_1 \) (let's say \( x_1 = 0 \) for simplicity).
- Mass \( m' \) is at position \( x_2 \) (we can assume \( x_2 = d \), where \( d \) is some distance).

Thus, the center of mass will initially be:
\[ x_{cm} = \frac{m \cdot 0 + m' \cdot d}{m + m'} = \frac{m' \cdot d}{m + m'} \]

3. **Moving Mass \( m \):**
When mass \( m \) is moved by a distance \( x \), its new position becomes \( x_1' = x \). The position of mass \( m' \) remains the same at \( x_2 = d \).

The new center of mass \( x_{cm}' \) is:
\[ x_{cm}' = \frac{m \cdot x + m' \cdot d}{m + m'} \]

4. **Change in Center of Mass:**
According to the problem, the center of mass moves by a distance \( \frac{x}{5} \):
\[ x_{cm}' - x_{cm} = \frac{x}{5} \]

5. **Substituting the Expressions:**
Substituting the expressions for \( x_{cm}' \) and \( x_{cm} \):
\[ \frac{m \cdot x + m' \cdot d}{m + m'} - \frac{m' \cdot d}{m + m'} = \frac{x}{5} \]

This simplifies to:
\[ \frac{m \cdot x}{m + m'} = \frac{x}{5} \]

6. **Cross Multiplying:**
Cross multiplying gives:
\[ 5m \cdot x = (m + m') \cdot x \]

Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 5m = m + m' \]

7. **Rearranging the Equation:**
Rearranging gives:
\[ 5m - m = m' \implies 4m = m' \]

8. **Finding the Ratio:**
Thus, the ratio \( \frac{m'}{m} \) is:
\[ \frac{m'}{m} = \frac{4m}{m} = 4 \]

### Final Answer:
The ratio \( \frac{m'}{m} \) is \( 4 \).