A pulley fixed to the ceiling carries a string with blocks of mass `m` and `3m` attached to its ends. The masses of string and pulley are negligible .When the system is released, its center of mass moves with what acceleration

Options

A. `0`
B. `g//4`
C. `g//2`
D. `-g//2`

Explanation

To find the acceleration of the center of mass of the system consisting of two blocks of masses `m` and `3m` connected by a string over a pulley, we can follow these steps:

### Step 1: Identify the forces acting on the system
When the system is released, the block of mass `3m` will accelerate downward due to gravity, while the block of mass `m` will accelerate upward. The gravitational force acting on each block is given by:
- For block `m`: \( F_1 = mg \) (upward)
- For block `3m`: \( F_2 = 3mg \) (downward)

### Step 2: Calculate the net force acting on the system
The net force acting on the system can be calculated by considering the difference between the downward force and the upward force:
\[ F_{net} = F_2 - F_1 = 3mg - mg = 2mg \]

### Step 3: Determine the total mass of the system
The total mass of the system is the sum of the masses of both blocks:
\[ M_{total} = m + 3m = 4m \]

### Step 4: Apply Newton's second law to find the acceleration
According to Newton's second law, the acceleration \( a \) of the center of mass can be calculated using the formula:
\[ a = \frac{F_{net}}{M_{total}} = \frac{2mg}{4m} \]
Simplifying this gives:
\[ a = \frac{2g}{4} = \frac{g}{2} \]

### Step 5: Determine the direction of acceleration
Since the net force is directed downward (due to the heavier block), the acceleration of the center of mass will also be downward. Therefore, we can express the acceleration as:
\[ a = \frac{g}{2} \text{ (downward)} \]

### Final Answer
The acceleration of the center of mass of the system is \( \frac{g}{2} \) downward.

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