A man pulls out a block of mass m from the depth d with help of a light string. If acceleration of the block is `g/2` then the work done by the string on the block will be

Options

A. `-3mgd /2`
B. `+3mgd /2`
C. `-mgd /2`
D. `+mgd /2`

Answer

To solve the problem of finding the work done by the string on the block, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Block:**
The block of mass \( m \) is subjected to two main forces:
- The gravitational force acting downward: \( F_g = mg \)
- The tension in the string acting upward: \( T \)

2. **Apply Newton's Second Law:**
According to Newton's second law, the net force acting on the block is equal to the mass of the block multiplied by its acceleration. The block is accelerating upwards with an acceleration of \( \frac{g}{2} \). Therefore, we can write the equation:
\[ T - mg = ma \]
where \( a = \frac{g}{2} \).

3. **Substitute the Values:**
Substituting \( a \) into the equation gives:
\[ T - mg = m \left(\frac{g}{2}\right) \]

4. **Rearranging the Equation:**
Rearranging the equation to solve for \( T \):
\[ T = mg + m \left(\frac{g}{2}\right) \]
\[ T = mg + \frac{mg}{2} \]
\[ T = \frac{2mg}{2} + \frac{mg}{2} = \frac{3mg}{2} \]

5. **Calculate the Work Done by the String:**
The work done \( W \) by the tension in the string as the block is pulled out from depth \( d \) is given by the formula:
\[ W = T \cdot d \]
Substituting the value of \( T \):
\[ W = \left(\frac{3mg}{2}\right) \cdot d \]
\[ W = \frac{3mgd}{2} \]

### Final Answer:
The work done by the string on the block is \( \frac{3mgd}{2} \).

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