The angle turned by a body undergoing circular motion depends on time as `theta = theta_(0)+theta_(1)t+theta_(2)t^(2)`. Then the angular acceleration of the body is

Options

A. `theta_(1)`
B. `theta_(2)`
C. `2 theta_(1)`
D. `2 theta_(2)`

Explanation

To find the angular acceleration of a body undergoing circular motion, given the angle turned as a function of time, we can follow these steps:

### Step 1: Write down the equation for angular displacement
The angle turned by the body is given by:
\[ \theta = \theta_0 + \theta_1 t + \theta_2 t^2 \]
where \(\theta_0\), \(\theta_1\), and \(\theta_2\) are constants.

### Step 2: Differentiate to find angular velocity
To find the angular velocity (\(\omega\)), we take the first derivative of \(\theta\) with respect to time \(t\):
\[ \frac{d\theta}{dt} = \frac{d}{dt}(\theta_0 + \theta_1 t + \theta_2 t^2) \]
Since \(\theta_0\) is a constant, its derivative is 0. Thus, we have:
\[ \frac{d\theta}{dt} = \theta_1 + 2\theta_2 t \]

### Step 3: Differentiate again to find angular acceleration
Now, we need to find the angular acceleration (\(\alpha\)), which is the second derivative of \(\theta\) with respect to time:
\[ \frac{d^2\theta}{dt^2} = \frac{d}{dt}(\theta_1 + 2\theta_2 t) \]
Again, since \(\theta_1\) is a constant, its derivative is 0. Therefore, we have:
\[ \frac{d^2\theta}{dt^2} = 2\theta_2 \]

### Step 4: Conclusion
The angular acceleration \(\alpha\) is given by:
\[ \alpha = \frac{d^2\theta}{dt^2} = 2\theta_2 \]

Thus, the angular acceleration of the body is \(2\theta_2\).

### Final Answer
The correct option is:
**Option 4: \(2\theta_2\)**

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