In which of the following cases will the work done be maximum? The body is moved through a distance S on the ground

Options

A.
B.
C.
D.

Answer

To determine in which case the work done is maximum when a body is moved through a distance \( S \) on the ground, we can use the formula for work done:

\[ W = F \cdot S \cdot \cos(\theta) \]

Where:
- \( W \) is the work done,
- \( F \) is the force applied,
- \( S \) is the distance moved,
- \( \theta \) is the angle between the force and the direction of displacement.

### Step-by-Step Solution:

1. **Identify the Formula**:
The work done by a force is given by the equation \( W = F \cdot S \cdot \cos(\theta) \).

2. **Understand the Role of \( \theta \)**:
The angle \( \theta \) affects the value of \( \cos(\theta) \). The maximum value of \( \cos(\theta) \) is 1, which occurs when \( \theta = 0^\circ \) (force and displacement are in the same direction).

3. **Analyze Different Cases**:
- **Case 1**: \( \theta = 30^\circ \)
\[ W_1 = F \cdot S \cdot \cos(30^\circ) = F \cdot S \cdot \frac{\sqrt{3}}{2} \]
- **Case 2**: \( \theta = 60^\circ \)
\[ W_2 = F \cdot S \cdot \cos(60^\circ) = F \cdot S \cdot \frac{1}{2} \]
- **Case 3**: \( \theta = 45^\circ \)
\[ W_3 = F \cdot S \cdot \cos(45^\circ) = F \cdot S \cdot \frac{1}{\sqrt{2}} \]
- **Case 4**: \( \theta = 0^\circ \)
\[ W_4 = F \cdot S \cdot \cos(0^\circ) = F \cdot S \cdot 1 \]

4. **Compare the Work Done**:
- From the calculations:
- \( W_1 = F \cdot S \cdot \frac{\sqrt{3}}{2} \)
- \( W_2 = F \cdot S \cdot \frac{1}{2} \)
- \( W_3 = F \cdot S \cdot \frac{1}{\sqrt{2}} \)
- \( W_4 = F \cdot S \cdot 1 \)

5. **Determine Maximum Work Done**:
Since \( \cos(\theta) \) decreases as \( \theta \) increases, the maximum work done occurs when \( \theta \) is at its minimum, which is \( 0^\circ \). Thus, the maximum work done is:
\[ W_4 = F \cdot S \]

### Conclusion:
The work done is maximum in the case where the angle \( \theta \) is \( 0^\circ \) (Case 4). Therefore, the answer to the question is **Option 4**.