A points is A located on the rim of a wheel of radius R which rolls without slipping along a horizontal surface with velocity v. The total distance travelled by the point A between successive moments at which it touches the surface is :

Options

A. 4R
B. 2R
C. 8R
D. `sqrt(8)R`

Answer

To solve the problem of finding the total distance traveled by point A on the rim of a wheel of radius R that rolls without slipping along a horizontal surface with velocity v, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion of the Wheel**:
- When the wheel rolls without slipping, the point A on the rim of the wheel will touch the ground and then move up and around the wheel as it rolls forward.

2. **Distance Traveled by the Wheel**:
- The distance traveled by the center of the wheel in one complete revolution is equal to the circumference of the wheel. The circumference \( C \) of a circle is given by:
\[ C = 2\pi R \]

3. **Finding the Distance Traveled by Point A**:
- As the wheel rolls, point A will touch the ground and then travel along the circular path of the wheel until it touches the ground again. During one complete revolution of the wheel, point A will travel a distance equal to the circumference of the wheel plus the distance it moves horizontally while the wheel rolls.
- The horizontal distance traveled by the center of the wheel during one complete revolution is \( 2\pi R \).

4. **Total Distance Traveled by Point A**:
- Since point A travels along the circular path and also moves horizontally, the total distance traveled by point A in one complete revolution is:
\[ \text{Total Distance} = \text{Circumference} + \text{Horizontal Distance} = 2\pi R + 2\pi R = 4\pi R \]

5. **Conclusion**:
- Therefore, the total distance traveled by point A between successive moments at which it touches the surface is:
\[ \text{Total Distance} = 4\pi R \]