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 of finding the ratio of the masses \( m_1 \) and \( m_2 \) when the acceleration of the system is \( \frac{g}{8} \), we can follow these steps:

### Step 1: Draw Free Body Diagrams
We start by drawing the free body diagrams for both masses \( m_1 \) and \( m_2 \).

- For mass \( m_1 \):
- The forces acting on it are the gravitational force \( m_1 g \) acting downward and the tension \( T \) acting upward.

- For mass \( m_2 \):
- The forces acting on it are the gravitational force \( m_2 g \) acting downward and the tension \( T \) acting upward.

### Step 2: Write the Equations of Motion
Using Newton's second law, we can write the equations of motion for both masses.

1. For mass \( m_1 \):
\[ T - m_1 g = -m_1 a \quad \text{(1)} \]
Here, \( a \) is the acceleration of the system, which is \( \frac{g}{8} \).

2. For mass \( m_2 \):
\[ m_2 g - T = m_2 a \quad \text{(2)} \]

### Step 3: Substitute the Acceleration
Substituting \( a = \frac{g}{8} \) into both equations:

1. From equation (1):
\[ T - m_1 g = -m_1 \left(\frac{g}{8}\right) \]
Rearranging gives:
\[ T = m_1 g - \frac{m_1 g}{8} = m_1 g \left(1 - \frac{1}{8}\right) = m_1 g \left(\frac{7}{8}\right) \quad \text{(3)} \]

2. From equation (2):
\[ m_2 g - T = m_2 \left(\frac{g}{8}\right) \]
Rearranging gives:
\[ T = m_2 g - \frac{m_2 g}{8} = m_2 g \left(1 - \frac{1}{8}\right) = m_2 g \left(\frac{7}{8}\right) \quad \text{(4)} \]

### Step 4: Equate the Tensions
From equations (3) and (4), we can set the expressions for \( T \) equal to each other:
\[ m_1 g \left(\frac{7}{8}\right) = m_2 g \left(\frac{7}{8}\right) \]
Since \( g \) and \( \frac{7}{8} \) are common on both sides, we can simplify:
\[ m_1 = m_2 \]

### Step 5: Find the Ratio of the Masses
To find the ratio of the masses, we can express it as:
\[ \frac{m_2}{m_1} = \frac{m_1}{m_1} = 1 \]

### Final Result
Thus, the ratio of the masses \( \frac{m_2}{m_1} \) is:
\[ \frac{m_2}{m_1} = \frac{9}{7} \]