The angular displacement of a body is given by `theta = 2t^(2) + 5t -3`. Find the value of the angular velocity and angular acceleration when t = 2s.

Explanation

To solve the problem, we need to find the angular velocity and angular acceleration from the given angular displacement function. The angular displacement is given by:

\[ \theta(t) = 2t^2 + 5t - 3 \]

### Step 1: Find Angular Velocity
Angular velocity (\(\omega\)) is defined as the rate of change of angular displacement with respect to time. Mathematically, it is represented as:

\[ \omega = \frac{d\theta}{dt} \]

Now, we differentiate \(\theta(t)\) with respect to \(t\):

\[ \frac{d\theta}{dt} = \frac{d}{dt}(2t^2 + 5t - 3) \]

Using the power rule of differentiation:

\[ \frac{d\theta}{dt} = 2 \cdot 2t^{2-1} + 5 \cdot 1t^{1-1} - 0 = 4t + 5 \]

Thus, the expression for angular velocity is:

\[ \omega(t) = 4t + 5 \]

### Step 2: Calculate Angular Velocity at \(t = 2s\)
Now, we substitute \(t = 2\) seconds into the angular velocity equation:

\[ \omega(2) = 4(2) + 5 = 8 + 5 = 13 \text{ rad/s} \]

### Step 3: Find Angular Acceleration
Angular acceleration (\(\alpha\)) is defined as the rate of change of angular velocity with respect to time. It is given by:

\[ \alpha = \frac{d\omega}{dt} \]

Now, we differentiate \(\omega(t)\) with respect to \(t\):

\[ \frac{d\omega}{dt} = \frac{d}{dt}(4t + 5) \]

Differentiating gives:

\[ \frac{d\omega}{dt} = 4 \]

Thus, the angular acceleration is:

\[ \alpha = 4 \text{ rad/s}^2 \]

### Summary of Results
- The angular velocity at \(t = 2\) seconds is \(13 \text{ rad/s}\).
- The angular acceleration is \(4 \text{ rad/s}^2\).