A particle is going parallel to x-axis with constant speed V at a distance 'a' from the axis. Find its angular velocity about an axis passing the origin O, at the instant when radial vector of the particle makes angle `theta` with the x-axis

Options

A. `(V)/(a)sin^(2)theta`
B. `(V)/(asin(theta)`
C. `(2V)/(a)sin^(2)theta`
D. `(V)/(2a)sin^(2)theta`

Answer

To solve the problem, we need to find the angular velocity of a particle moving parallel to the x-axis at a constant speed \( V \) and at a distance \( a \) from the x-axis, when the radial vector makes an angle \( \theta \) with the x-axis.

### Step-by-Step Solution:

1. **Understand the Setup**:
- The particle is moving parallel to the x-axis with a constant speed \( V \).
- It is located at a distance \( a \) from the x-axis.
- The radial vector from the origin (O) to the particle makes an angle \( \theta \) with the x-axis.

2. **Identify the Position of the Particle**:
- The position of the particle can be represented in Cartesian coordinates as \( (x, y) \).
- Since it is moving parallel to the x-axis and at a distance \( a \) from the x-axis, we can denote the position as \( (x, a) \).

3. **Determine the Radial Vector**:
- The radial vector \( \vec{r} \) from the origin to the particle is given by:
\[ \vec{r} = x \hat{i} + a \hat{j} \]
- The angle \( \theta \) is the angle between this radial vector and the x-axis.

4. **Calculate the Perpendicular Distance**:
- The perpendicular distance from the axis of rotation (the x-axis) to the line of motion of the particle is given by:
\[ d = a \sin(\theta) \]
- This is because the radial vector makes an angle \( \theta \) with the x-axis, and the vertical component of the distance is \( a \sin(\theta) \).

5. **Determine the Angular Velocity**:
- The angular velocity \( \omega \) can be calculated using the formula:
\[ \omega = \frac{V}{d} \]
- Substituting the expression for \( d \):
\[ \omega = \frac{V}{a \sin(\theta)} \]

### Final Answer:
The angular velocity \( \omega \) of the particle about the axis passing through the origin is:
\[ \omega = \frac{V}{a \sin(\theta)} \]