If the equation for the displacement of a particle moving in a circular path is given by `(theta)=2t^(3)+0.5`, where `theta` is in radians and `t` in seconds, then the angular velocity of particle after `2 s` from its start is

Options

A. 8 rad/s
B. 12 rad/s
C. 24 rad/s
D. 36 rad/s

Answer

To find the angular velocity of a particle moving in a circular path given the equation for its angular displacement, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The angular displacement \(\theta\) is given by the equation:
\[ \theta = 2t^3 + 0.5 \]
where \(\theta\) is in radians and \(t\) is in seconds.

2. **Identify Angular Velocity**:
Angular velocity (\(\omega\)) is defined as the rate of change of angular displacement with respect to time. Mathematically, it can be expressed as:
\[ \omega = \frac{d\theta}{dt} \]

3. **Differentiate the Displacement Equation**:
To find \(\omega\), we need to differentiate \(\theta\) with respect to \(t\):
\[ \frac{d\theta}{dt} = \frac{d}{dt}(2t^3 + 0.5) \]
Using the power rule of differentiation:
\[ \frac{d\theta}{dt} = 6t^2 \]

4. **Calculate Angular Velocity at \(t = 2\) seconds**:
Now, we substitute \(t = 2\) seconds into the differentiated equation to find the angular velocity:
\[ \omega = 6(2^2) = 6 \times 4 = 24 \text{ radians/second} \]

5. **Conclusion**:
The angular velocity of the particle after \(2\) seconds is:
\[ \omega = 24 \text{ radians/second} \]