Two bodies with masses `m_1` and `m_2(m_1gtm_2)` are joined by a string passing over fixed pulley. Assume masses of the pulley and thread negligible. Then the acceleration of the centre of mass of the system `(m_1+m_2)` is

Options

A. (a) `((m_1-m_2)/(m_1+m_2))g`
B. (b) `((m_1-m_2)/(m_1+m_2))^2g`
C. (c) `(m_1g)/(m_1+m_2)`
D. (d) `(m_2g)/(m_1+m_2)`

Explanation

To find the acceleration of the center of mass of the system consisting of two bodies with masses \( m_1 \) and \( m_2 \) (where \( m_1 > m_2 \)), we can follow these steps:

### Step 1: Identify the Forces Acting on the Masses
- The weight of mass \( m_1 \) acting downward is \( m_1 g \).
- The weight of mass \( m_2 \) acting downward is \( m_2 g \).
- The tension in the string acts upward on both masses.

### Step 2: Write the Net Force Equation
The net force acting on the system can be expressed as:
\[ F_{\text{net}} = m_1 g - m_2 g \]
This is because \( m_1 \) is heavier and will accelerate downward.

### Step 3: Simplify the Net Force
We can factor out \( g \):
\[ F_{\text{net}} = g(m_1 - m_2) \]

### Step 4: Determine the Total Mass of the System
The total mass \( M \) of the system is:
\[ M = m_1 + m_2 \]

### Step 5: Apply Newton's Second Law
According to Newton's second law, the net force is equal to the total mass times the acceleration \( a \):
\[ F_{\text{net}} = M a \]
Substituting the expressions we have:
\[ g(m_1 - m_2) = (m_1 + m_2) a \]

### Step 6: Solve for Acceleration \( a \)
Rearranging the equation to solve for \( a \):
\[ a = \frac{g(m_1 - m_2)}{m_1 + m_2} \]

### Step 7: Find the Acceleration of the Center of Mass
The acceleration of the center of mass \( a_{CM} \) is the same as the acceleration \( a \) we derived:
\[ a_{CM} = \frac{g(m_1 - m_2)}{m_1 + m_2} \]

### Final Answer
Thus, the acceleration of the center of mass of the system is:
\[ a_{CM} = \frac{g(m_1 - m_2)}{m_1 + m_2} \]