A body of mass m is allowed to fall with the help of string with downward acceleration `(g)/(6)` to a distance x. The work done by the string is

Options

A. `(mgx)/(6)`
B. `-(mgx)/(6)`
C. `(5mgx)/(6)`
D. `-(5mgx)/(6)`

Answer

To solve the problem, we need to find the work done by the string when a body of mass \( m \) falls with a downward acceleration of \( \frac{g}{6} \) over a distance \( x \).

### Step-by-step Solution:

1. **Identify the Forces Acting on the Mass**:
- The weight of the mass \( m \) acting downwards: \( F_g = mg \).
- The tension \( T \) in the string acting upwards.

2. **Apply Newton's Second Law**:
- According to Newton's second law, the net force acting on the mass is equal to the mass times its acceleration:
\[ F_{\text{net}} = ma \]
- Here, the net force can be expressed as the difference between the gravitational force and the tension:
\[ mg - T = ma \]
- Given that the downward acceleration \( a = \frac{g}{6} \), we can substitute this into the equation:
\[ mg - T = m \left(\frac{g}{6}\right) \]

3. **Rearranging the Equation**:
- Rearranging the above equation to solve for tension \( T \):
\[ T = mg - m \left(\frac{g}{6}\right) \]
- Simplifying this gives:
\[ T = mg - \frac{mg}{6} = mg \left(1 - \frac{1}{6}\right) = mg \left(\frac{5}{6}\right) \]
- Therefore, the tension in the string is:
\[ T = \frac{5mg}{6} \]

4. **Calculate the Work Done by the String**:
- The work done by the tension in the string is given by the formula:
\[ W = F \cdot d \cdot \cos(\theta) \]
- Here, the force \( F \) is the tension \( T \), the displacement \( d \) is \( x \), and the angle \( \theta \) between the force and displacement is \( 180^\circ \) (since tension acts upwards while displacement is downwards).
- Thus, we have:
\[ W = T \cdot x \cdot \cos(180^\circ) = -T \cdot x \]
- Substituting the value of \( T \):
\[ W = -\left(\frac{5mg}{6}\right) \cdot x \]
- Therefore, the work done by the string is:
\[ W = -\frac{5mgx}{6} \]

### Final Answer:
The work done by the string is:
\[ W = -\frac{5mgx}{6} \]