A particle undergoes uniform circular motion. The velocity and angular velocity of the particle at an instant of time is `v=3hat i+4hatj m//s` and `vec(omega )= xhat i+6hatj rad//sec`
The value of x in rad/s is

Options

A. 8
B. -8
C. 6
D. can't be calculated

Answer

To solve the problem, we need to find the value of \( x \) in the angular velocity vector \( \vec{\omega} = x \hat{i} + 6 \hat{j} \) given the velocity vector \( \vec{v} = 3 \hat{i} + 4 \hat{j} \).

### Step-by-Step Solution:

1. **Understand the relationship between velocity and angular velocity**:
For a particle in uniform circular motion, the velocity vector \( \vec{v} \) and the angular velocity vector \( \vec{\omega} \) are perpendicular to each other. This means their dot product is zero.

2. **Set up the dot product**:
The dot product of \( \vec{v} \) and \( \vec{\omega} \) can be expressed as:
\[ \vec{v} \cdot \vec{\omega} = (3 \hat{i} + 4 \hat{j}) \cdot (x \hat{i} + 6 \hat{j}) \]

3. **Calculate the dot product**:
Using the properties of the dot product:
\[ \vec{v} \cdot \vec{\omega} = 3x + 4 \cdot 6 \]
This simplifies to:
\[ 3x + 24 \]

4. **Set the dot product equal to zero**:
Since the vectors are perpendicular:
\[ 3x + 24 = 0 \]

5. **Solve for \( x \)**:
Rearranging the equation gives:
\[ 3x = -24 \]
Dividing both sides by 3:
\[ x = -8 \]

6. **Conclusion**:
The value of \( x \) in radians per second is:
\[ x = -8 \, \text{rad/s} \]