A wheel of radius `1 m` rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially on contact with the ground is.

Options

A. `2 pi`
B. `sqrt(2)pi`
C. `sqrt(pi^(2)+4)`
D. `pi`

Explanation

To solve the problem, we need to find the magnitude of the displacement of the point on the wheel that was initially in contact with the ground after the wheel rolls forward half a revolution.

### Step-by-step Solution:

1. **Identify the Radius and Circumference of the Wheel**:
- Given the radius \( r = 1 \, \text{m} \).
- The circumference \( C \) of the wheel is calculated using the formula:
\[ C = 2 \pi r = 2 \pi \times 1 = 2 \pi \, \text{m} \]

2. **Calculate the Distance Rolled for Half a Revolution**:
- For half a revolution, the distance \( d \) rolled by the wheel is:
\[ d = \frac{1}{2} \times C = \frac{1}{2} \times 2 \pi = \pi \, \text{m} \]

3. **Determine the Position of the Point After Half a Revolution**:
- Initially, let’s denote the point on the wheel that is in contact with the ground as point \( A \).
- After rolling half a revolution, this point will move to a new position \( A' \) directly above the center of the wheel.

4. **Calculate the Vertical Displacement**:
- The vertical distance from the ground to the center of the wheel is equal to the radius \( r = 1 \, \text{m} \).
- After half a revolution, the point \( A' \) will be at a height of \( 2 \, \text{m} \) (the diameter of the wheel) above the ground.

5. **Determine the Displacement**:
- The horizontal displacement \( AB \) is equal to the distance rolled, which is \( \pi \, \text{m} \).
- The vertical displacement \( A'B \) is equal to the diameter of the wheel, which is \( 2 \, \text{m} \).

6. **Use the Pythagorean Theorem to Find the Magnitude of the Displacement**:
- We can find the magnitude of the displacement \( AA' \) using the Pythagorean theorem:
\[ AA' = \sqrt{(AB)^2 + (A'B)^2} = \sqrt{(\pi)^2 + (2)^2} \]
- Simplifying this gives:
\[ AA' = \sqrt{\pi^2 + 4} \]

### Final Answer:
The magnitude of the displacement of the point of the wheel initially in contact with the ground after rolling forward half a revolution is:
\[ \sqrt{\pi^2 + 4} \, \text{m} \]