A disc rolls over a horizontal floor without slipping with a linear speed of 5 cm/sec. Then the linear speed of a particle on its rim, with respect to the floor, when it is in its highest position is?

Options

A. 10cm/sec
B. 5cm/sec
C. 2.5cm/sec
D. 0

Explanation

To solve the problem, we need to determine the linear speed of a particle on the rim of a disc when it is at its highest position, given that the disc rolls without slipping with a linear speed of 5 cm/sec.

### Step-by-Step Solution:

1. **Understand the Motion of the Disc**:
- The disc rolls on a horizontal surface without slipping. This means that the point of contact with the ground (let's call it point A) is momentarily at rest.

2. **Identify the Linear Speed**:
- The linear speed (V) of the center of the disc is given as 5 cm/sec.

3. **Relate Linear Speed to Angular Speed**:
- For a disc rolling without slipping, the relationship between linear speed (V) and angular speed (ω) is given by:
\[ V = \omega R \]
- Here, R is the radius of the disc.

4. **Determine the Speed at the Highest Point**:
- The highest point of the disc (let's call it point C) will have two components of velocity:
- The linear speed of the center of the disc (V) acting forward.
- The tangential speed due to rotation (ωR), which is also acting forward at the highest point.
- Since the disc rolls without slipping, at the highest point, the total speed (V_C) can be expressed as:
\[ V_C = V + \omega R \]
- Since \(V = \omega R\), we can substitute this into the equation:
\[ V_C = V + V = 2V \]

5. **Calculate the Speed at the Highest Point**:
- Now substituting the value of V:
\[ V_C = 2 \times 5 \text{ cm/sec} = 10 \text{ cm/sec} \]

6. **Conclusion**:
- The linear speed of the particle on the rim at its highest position with respect to the floor is **10 cm/sec**.