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. `vx`
D. `vr`

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 uniform angular speed \( x \). This means that the angular speed does not change over time.

2. **Identify Types of Acceleration**: In circular motion, there are two types of acceleration:
- **Tangential Acceleration (\( a_t \))**: This is due to the change in the speed of the particle along the circular path.
- **Radial Acceleration (\( a_r \))**: This is directed towards the center of the circular path and is responsible for changing the direction of the velocity vector.

3. **Determine Tangential Acceleration**: Since the particle is moving with uniform angular speed, the linear speed \( V \) is constant. Therefore, the tangential acceleration \( a_t \) is:
\[ a_t = \frac{dV}{dt} = 0 \]

4. **Calculate Radial Acceleration**: The radial acceleration can be calculated using the formula:
\[ a_r = \frac{V^2}{r} \quad \text{or} \quad a_r = r \omega^2 \]
where \( \omega \) is the angular speed. In this case, \( \omega = x \).

5. **Relate Linear Velocity to Angular Speed**: The linear velocity \( V \) can be expressed in terms of the radius \( r \) and angular speed \( x \):
\[ V = r \cdot x \]

6. **Substitute for Radial Acceleration**: Using the relationship \( V = r \cdot x \) in the radial acceleration formula:
\[ a_r = \frac{(r \cdot x)^2}{r} = \frac{r^2 \cdot x^2}{r} = r \cdot x^2 \]

7. **Final Result**: Since the tangential acceleration is zero, the total acceleration \( a \) of the particle is equal to the radial acceleration:
\[ a = a_r = r \cdot x^2 \]

### Conclusion:
The acceleration of the particle moving around a circular path with uniform angular speed \( x \) and radius \( r \) is given by:
\[ \text{Acceleration} = r \cdot x^2 \]