A Wheel of circumference C is at rest on the ground.When the wheel rolls forward through half a revolution ,then the displacement of point contact will be

Options

A. `Csqrt((1)/(pi^(2))+(1)/(4))`
B. `(C )/(2)`
C. `pisqrt(c^(2)+4)`
D. `Csqrt((1)/(pi)+(1)/(2))`

Answer

To solve the problem, we need to determine the displacement of the point of contact when a wheel rolls forward through half a revolution. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Wheel's Movement**:
- The wheel has a circumference \( C \).
- When the wheel rolls forward through half a revolution, it covers a distance equal to half of its circumference.

2. **Calculate the Distance Covered**:
- The distance covered by the wheel after half a revolution is given by:
\[ \text{Distance} = \frac{C}{2} \]

3. **Identify the Initial and Final Positions**:
- Let’s denote the initial point of contact on the ground as point A.
- After rolling half a revolution, the point of contact moves to point B.

4. **Determine the Vertical Displacement**:
- The radius of the wheel is denoted as \( r \).
- The vertical distance from point A to the center of the wheel is \( r \).
- After half a revolution, the center of the wheel moves horizontally by \( \frac{C}{2} \) and remains at the same vertical level.

5. **Use the Pythagorean Theorem**:
- The displacement \( s \) from point A to point B can be calculated using the Pythagorean theorem:
\[ s = \sqrt{\left(\frac{C}{2}\right)^2 + (2r)^2} \]
- Here, \( 2r \) is the vertical distance since the wheel rolls down to the ground level.

6. **Substituting the Values**:
- Substitute \( C = 2\pi r \) into the equation:
\[ s = \sqrt{\left(\frac{2\pi r}{2}\right)^2 + (2r)^2} \]
- Simplifying gives:
\[ s = \sqrt{\left(\pi r\right)^2 + (2r)^2} \]
- This simplifies to:
\[ s = \sqrt{\pi^2 r^2 + 4r^2} \]
- Factoring out \( r^2 \):
\[ s = r \sqrt{\pi^2 + 4} \]

7. **Final Expression for Displacement**:
- Thus, the displacement of the point of contact after rolling half a revolution is:
\[ s = r \sqrt{\pi^2 + 4} \]