A particle is moving around a circular path with uniform angular speed (x) . The radius of the circular path is (r). The acceleration of the particle is:

Options

A. `(x2)/(r )`
B. `(x)/(r )`
C. `rx^2`
D. `r/x`

Answer

To find the acceleration of a particle moving in a circular path with uniform angular speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**: The particle is moving in a circular path with a uniform angular speed \( \omega \) (given as \( x \)) and a radius \( r \).

2. **Identify Types of Acceleration**: In circular motion, the total acceleration of the particle can be broken down into two components:
- Tangential acceleration (\( a_t \)): This is due to the change in the speed of the particle along the circular path.
- Centripetal acceleration (\( a_c \)): This is due to the change in direction of the velocity vector as the particle moves along the circular path.

3. **Determine Tangential Acceleration**: Since the particle is moving with uniform angular speed, its speed does not change. Therefore, the tangential acceleration is:
\[ a_t = 0 \]

4. **Calculate Centripetal Acceleration**: The centripetal acceleration can be calculated using the formula:
\[ a_c = \omega^2 r \]
where \( \omega \) is the angular speed and \( r \) is the radius of the circular path.

5. **Substitute the Given Values**: We know that \( \omega = x \). Substituting this into the centripetal acceleration formula gives:
\[ a_c = x^2 r \]

6. **Conclusion**: Since the tangential acceleration is zero, the total acceleration of the particle is equal to the centripetal acceleration:
\[ a = a_c = x^2 r \]

### Final Answer:
The acceleration of the particle is \( x^2 r \).