A light string passing over a smooth light pulley connects two blocks of masses `m_1` and `m_2` (vertically). If the acceleration of the system is `g//8`, then the ratio of the masses is

Options

A. (a) 8:1
B. (b) 9:7
C. (c) 4:3
D. (d) 5:3

Explanation

To solve the problem, we need to analyze the forces acting on the two masses connected by a string over a pulley and use Newton's second law of motion.

### Step-by-Step Solution:

1. **Identify the Forces**:
- For mass \( m_1 \) (which is moving downwards), the forces acting on it are:
- Weight: \( m_1 g \) (downwards)
- Tension: \( T \) (upwards)
- For mass \( m_2 \) (which is moving upwards), the forces acting on it are:
- Weight: \( m_2 g \) (downwards)
- Tension: \( T \) (upwards)

2. **Write the Equations of Motion**:
- For mass \( m_1 \):
\[ m_1 g - T = m_1 a \quad \text{(1)} \]
- For mass \( m_2 \):
\[ T - m_2 g = m_2 a \quad \text{(2)} \]

3. **Substitute the Given Acceleration**:
- We know that the acceleration \( a = \frac{g}{8} \). Substitute this into equations (1) and (2):
- For equation (1):
\[ m_1 g - T = m_1 \left(\frac{g}{8}\right) \]
Rearranging gives:
\[ T = m_1 g - m_1 \left(\frac{g}{8}\right) = m_1 g \left(1 - \frac{1}{8}\right) = m_1 g \left(\frac{7}{8}\right) \quad \text{(3)} \]
- For equation (2):
\[ T - m_2 g = m_2 \left(\frac{g}{8}\right) \]
Rearranging gives:
\[ T = m_2 g + m_2 \left(\frac{g}{8}\right) = m_2 g \left(1 + \frac{1}{8}\right) = m_2 g \left(\frac{9}{8}\right) \quad \text{(4)} \]

4. **Set Equations (3) and (4) Equal**:
- Since both expressions equal \( T \), we can set them equal to each other:
\[ m_1 g \left(\frac{7}{8}\right) = m_2 g \left(\frac{9}{8}\right) \]

5. **Cancel \( g \) from Both Sides**:
- \( g \) cancels out from both sides:
\[ m_1 \left(\frac{7}{8}\right) = m_2 \left(\frac{9}{8}\right) \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ \frac{m_1}{m_2} = \frac{9}{7} \]

7. **Conclusion**:
- The ratio of the masses \( m_1 \) to \( m_2 \) is:
\[ \frac{m_1}{m_2} = \frac{9}{7} \]

### Final Answer:
The ratio of the masses \( m_1 \) and \( m_2 \) is \( 9:7 \).

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