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. `8:1`
B. `9:7`
C. `4:3`
D. `5:3`

Answer

To solve the problem, we need to analyze the forces acting on the two masses connected by a light string over a smooth pulley. Let's denote the masses as \( m_1 \) and \( m_2 \), and the acceleration of the system is given as \( \frac{g}{8} \).

### Step 1: Identify the forces acting on the masses
- For mass \( m_1 \), the force acting downwards is \( m_1 g \) and the tension \( T \) acts upwards.
- For mass \( m_2 \), the force acting downwards is \( m_2 g \) and the tension \( T \) acts upwards.

### Step 2: Write the equations of motion
Since the system is accelerating, we can write the equations of motion for both masses.

For \( m_1 \):
\[ m_1 g - T = m_1 a \quad \text{(1)} \]

For \( m_2 \):
\[ T - m_2 g = -m_2 a \quad \text{(2)} \]
(Note: The negative sign indicates that \( m_2 \) is accelerating downwards.)

### Step 3: Substitute the acceleration
Given that \( a = \frac{g}{8} \), we can substitute this into equations (1) and (2).

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

From equation (2):
\[ T - m_2 g = -m_2 \left(\frac{g}{8}\right) \]
Rearranging gives:
\[ T = m_2 g - m_2 \left(\frac{g}{8}\right) = m_2 g \left(1 + \frac{1}{8}\right) = m_2 g \left(\frac{9}{8}\right) \quad \text{(4)} \]

### Step 4: Equate the two expressions for tension
From equations (3) and (4), we have:
\[ m_1 g \left(\frac{7}{8}\right) = m_2 g \left(\frac{9}{8}\right) \]
We can cancel \( g \) from both sides (assuming \( g \neq 0 \)):
\[ m_1 \left(\frac{7}{8}\right) = m_2 \left(\frac{9}{8}\right) \]

### Step 5: Solve for the ratio of the masses
Rearranging gives:
\[ \frac{m_2}{m_1} = \frac{7}{9} \]

### Step 6: Write the final ratio
Thus, the ratio of the masses \( m_2 : m_1 \) is:
\[ m_2 : m_1 = 9 : 7 \]

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

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