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

Options

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

Explanation

To solve the problem of calculating the work done by the string on a body of mass \( m \) that is falling with a downward acceleration of \( g/6 \) over a distance \( x \), we can follow these steps:

### Step 1: Identify the Forces Acting on the Body
The forces acting on the body are:
- The gravitational force \( F_g = mg \) acting downward.
- The tension in the string \( T \) acting upward.

### Step 2: Apply Newton's Second Law
According to Newton's second law, the net force \( F_{net} \) acting on the body can be expressed as:
\[ F_{net} = m \cdot a \]
where \( a \) is the acceleration of the body. In this case, the body is accelerating downward with an acceleration of \( g/6 \).

### Step 3: Set Up the Equation for Net Force
The net force can also be expressed as the difference between the gravitational force and the tension:
\[ F_{net} = mg - T \]
Setting the two expressions for net force equal gives us:
\[ mg - T = m \cdot \frac{g}{6} \]

### Step 4: Solve for Tension \( T \)
Rearranging the equation to solve for \( T \):
\[ T = mg - m \cdot \frac{g}{6} \]
\[ T = mg \left(1 - \frac{1}{6}\right) = mg \cdot \frac{5}{6} \]
Thus, the tension in the string is:
\[ T = \frac{5mg}{6} \]

### Step 5: Calculate the Work Done by the String
The work done \( W \) by the tension in the string is given by:
\[ W = T \cdot d \cdot \cos(\theta) \]
where \( d \) is the distance moved by the body and \( \theta \) is the angle between the tension and the direction of displacement. Since the tension acts upward and the displacement is downward, \( \theta = 180^\circ \), and \( \cos(180^\circ) = -1 \).

Substituting the values:
\[ W = T \cdot x \cdot (-1) \]
\[ W = -T \cdot x \]
Substituting the value of \( T \):
\[ W = -\left(\frac{5mg}{6}\right) \cdot x \]
Thus, the work done by the string is:
\[ W = -\frac{5mgx}{6} \]

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