To solve the problem, we need to determine the displacement of point P, which is the contact point of a wheel with the ground, after the wheel has completed half of a rotation. Given that the radius of the wheel is \( r = 1 \, \text{m} \), we can follow these steps:
### Step-by-Step Solution:
1. **Understand the Movement of the Wheel**:
When the wheel rolls without slipping, the point of contact with the ground moves along with the wheel. After half a rotation, the point that was initially in contact with the ground will have moved to a new position.
2. **Calculate the Distance Traveled by the Wheel**:
The distance traveled by the wheel after half a rotation is equal to half of the circumference of the wheel. The circumference \( C \) of a circle is given by:
\[
C = 2\pi r
\]
For a radius \( r = 1 \, \text{m} \):
\[
C = 2\pi \times 1 = 2\pi \, \text{m}
\]
Therefore, the distance traveled after half a rotation is:
\[
\text{Distance} = \frac{1}{2} C = \frac{1}{2} \times 2\pi = \pi \, \text{m}
\]
3. **Determine the Final Position of Point P**:
Initially, point P is at the bottom of the wheel (the contact point). After half a rotation, point P will have moved to the opposite side of the wheel, which is directly above the original contact point. The vertical distance from the ground to the top of the wheel is equal to the radius of the wheel, which is \( 1 \, \text{m} \).
4. **Calculate the Displacement**:
The displacement of point P is the straight-line distance from its initial position (at the bottom) to its final position (at the top). This can be visualized as a right triangle where:
- One leg is the distance traveled horizontally (which is \( \pi \, \text{m} \)),
- The other leg is the vertical distance (which is \( 1 \, \text{m} \)).
Using the Pythagorean theorem to find the displacement \( d \):
\[
d = \sqrt{(\text{horizontal distance})^2 + (\text{vertical distance})^2}
\]
\[
d = \sqrt{(\pi)^2 + (1)^2}
\]
\[
d = \sqrt{\pi^2 + 1}
\]
5. **Final Result**:
The displacement of point P after the wheel completes half a rotation is:
\[
d = \sqrt{\pi^2 + 1} \, \text{m}
\]
