A lawn roller is pulled along a horizontal surface through a distance of `20 m` by a rope with a force of `200 N`. If the rope makes an angle of `60^@` with the vertical while pulling, the amount of work done by pulling force is

Options

A. 4000 J
B. 1000 J
C. `2000 sqrt(3) J`
D. 2000 J

Explanation

To solve the problem, we need to calculate the work done by the pulling force on the lawn roller. The work done (W) is given by the formula:

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

where:
- \( F \) is the force applied,
- \( d \) is the distance moved,
- \( \theta \) is the angle between the force and the direction of motion.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Force \( F = 200 \, \text{N} \)
- Distance \( d = 20 \, \text{m} \)
- Angle \( \theta = 60^\circ \) (with the vertical)

2. **Determine the Angle with the Horizontal:**
Since the angle is given with the vertical, we need to find the angle with the horizontal. The angle with the horizontal will be:
\[ \theta_{\text{horizontal}} = 90^\circ - 60^\circ = 30^\circ \]

3. **Calculate the Work Done:**
Using the formula for work done:
\[ W = F \cdot d \cdot \cos(\theta_{\text{horizontal}}) \]
Substitute the values:
\[ W = 200 \, \text{N} \cdot 20 \, \text{m} \cdot \cos(30^\circ) \]

4. **Calculate \( \cos(30^\circ) \):**
The cosine of 30 degrees is:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

5. **Substitute \( \cos(30^\circ) \) into the Work Done Equation:**
\[ W = 200 \cdot 20 \cdot \frac{\sqrt{3}}{2} \]

6. **Simplify the Equation:**
\[ W = 200 \cdot 20 \cdot \frac{\sqrt{3}}{2} = 200 \cdot 10 \cdot \sqrt{3} = 2000\sqrt{3} \, \text{J} \]

7. **Final Answer:**
The amount of work done by the pulling force is:
\[ W = 2000\sqrt{3} \, \text{J} \]