A point P on the rim of wheel is intially at rest and in contact with the ground. Find the displacement of the point P if the radius of the wheel is 5 m and the wheel rolls forward through half a revolution

Options

A. 5m
B. 10 m
C. `2.5` m
D. `5(sqrt((pi^(2)+4)))m`

Explanation

To solve the problem, we need to find the displacement of point P on the rim of a wheel that rolls forward through half a revolution, given that the radius of the wheel is 5 meters.

### Step-by-Step Solution:

1. **Understand the Initial Position**:
- The point P is initially at rest and in contact with the ground. This means it is at the bottom of the wheel.

2. **Determine the Radius**:
- The radius of the wheel (r) is given as 5 meters.

3. **Calculate the Diameter**:
- The diameter (D) of the wheel is twice the radius:
\[ D = 2r = 2 \times 5 = 10 \text{ meters} \]

4. **Identify the Movement**:
- When the wheel rolls forward through half a revolution, point P moves from the bottom of the wheel to the top of the wheel.

5. **Calculate the Linear Distance Traveled by the Center of Mass**:
- During half a revolution, the center of mass of the wheel moves forward by a distance equal to half the circumference of the wheel:
\[ \text{Circumference} = 2\pi r = 2\pi \times 5 = 10\pi \text{ meters} \]
\[ \text{Distance traveled} = \frac{1}{2} \times 10\pi = 5\pi \text{ meters} \]

6. **Determine the Final Position of Point P**:
- After half a revolution, point P is now at the top of the wheel. The center of mass has moved forward by \(5\pi\) meters, and point P is now vertically above the center of mass at a height equal to the radius (5 meters).

7. **Use the Pythagorean Theorem to Find Displacement**:
- The displacement (S) is the straight-line distance from the initial position of point P (at the bottom) to its final position (at the top). This forms a right triangle where:
- One leg is the vertical distance (the radius) = 5 meters.
- The other leg is the horizontal distance (distance traveled by the center of mass) = \(5\pi\) meters.
- Using the Pythagorean theorem:
\[ S = \sqrt{(5\pi)^2 + (5)^2} \]
\[ S = \sqrt{25\pi^2 + 25} = \sqrt{25(\pi^2 + 1)} = 5\sqrt{\pi^2 + 1} \]

8. **Final Calculation**:
- Substitute the value of \(r\) (5 meters) into the displacement formula:
\[ S = 5\sqrt{\pi^2 + 1} \text{ meters} \]

### Final Answer:
The displacement of point P after the wheel rolls forward through half a revolution is:
\[ S = 5\sqrt{\pi^2 + 1} \text{ meters} \]