Angular displacement `(theta)` of a flywheel varies with time as `theta = at+bt^(2)+ct^(3)` then angular acceleration is given by

Options

A. `a+2bt-3ct^(2)`
B. `2b-6t`
C. `a+2b-6t`
D. `2b+6ct`

Explanation

To find the angular acceleration of the flywheel given the angular displacement equation \( \theta = at + bt^2 + ct^3 \), we can follow these steps:

### Step 1: Understand the given equation
The angular displacement \( \theta \) is given as a function of time \( t \):
\[ \theta(t) = at + bt^2 + ct^3 \]
where \( a \), \( b \), and \( c \) are constants.

### Step 2: Find the angular velocity
Angular velocity \( \omega \) is the first derivative of angular displacement with respect to time:
\[ \omega = \frac{d\theta}{dt} \]
Calculating this derivative:
\[ \omega = \frac{d}{dt}(at + bt^2 + ct^3) = a + 2bt + 3ct^2 \]

### Step 3: Find the angular acceleration
Angular acceleration \( \alpha \) is the derivative of angular velocity with respect to time:
\[ \alpha = \frac{d\omega}{dt} \]
Calculating this derivative:
\[ \alpha = \frac{d}{dt}(a + 2bt + 3ct^2) = 0 + 2b + 6ct \]
Thus, the angular acceleration is:
\[ \alpha = 2b + 6ct \]

### Final Answer
The angular acceleration of the flywheel is given by:
\[ \alpha = 2b + 6ct \]

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