If angular velocity of a disc depends an angle rotated `theta` as `omega=theta^(2)+2theta`, then its angular acceleration `alpha` at `theta=1` rad is :

Options

A. `8 rad//s^(2)`
B. `10 rad//s^(2)`
C. `12 rad//s^(2)`
D. None of these

Explanation

To find the angular acceleration \( \alpha \) at \( \theta = 1 \) rad, we start with the given angular velocity equation:

\[ \omega = \theta^2 + 2\theta \]

### Step 1: Find \( \frac{d\omega}{d\theta} \)

To find the angular acceleration \( \alpha \), we first need to calculate the derivative of \( \omega \) with respect to \( \theta \):

\[ \frac{d\omega}{d\theta} = \frac{d}{d\theta}(\theta^2 + 2\theta) \]

Using basic differentiation rules:

\[ \frac{d\omega}{d\theta} = 2\theta + 2 \]

### Step 2: Substitute \( \theta = 1 \)

Now we substitute \( \theta = 1 \) rad into the derivative we just calculated:

\[ \frac{d\omega}{d\theta} \bigg|_{\theta=1} = 2(1) + 2 = 2 + 2 = 4 \]

### Step 3: Calculate \( \omega \) at \( \theta = 1 \)

Next, we calculate \( \omega \) at \( \theta = 1 \):

\[ \omega \bigg|_{\theta=1} = 1^2 + 2(1) = 1 + 2 = 3 \]

### Step 4: Calculate Angular Acceleration \( \alpha \)

Now we can find the angular acceleration \( \alpha \) using the formula:

\[ \alpha = \omega \cdot \frac{d\omega}{d\theta} \]

Substituting the values we found:

\[ \alpha = 3 \cdot 4 = 12 \, \text{rad/s}^2 \]

### Final Answer

Thus, the angular acceleration \( \alpha \) at \( \theta = 1 \) rad is:

\[ \alpha = 12 \, \text{rad/s}^2 \]

---