A particle in rotating disc covers angle `theta= 2t-t^(3)`

Options

A. Average angular velocity is `2 rad//s`
B. Average angular acceleration during first second is `- 4 rad//s^(2)`
C. Angular acceleration at the end of first second -` 6 rad//s`
D. Average angular acceleration during first second is -`6 rad//s^(2)`

Answer

To solve the problem step by step, we will follow the process of finding angular velocity and angular acceleration from the given angle function.

### Step 1: Understand the given function
The angle covered by the particle is given by:
\[ \theta(t) = 2t - t^3 \]

### Step 2: Find angular velocity
Angular velocity (\(\omega\)) is defined as the derivative of the angle with respect to time:
\[ \omega(t) = \frac{d\theta}{dt} \]
Calculating the derivative:
\[ \omega(t) = \frac{d}{dt}(2t - t^3) = 2 - 3t^2 \]

### Step 3: Calculate angular velocity at \(t = 0\)
Substituting \(t = 0\) into the angular velocity equation:
\[ \omega(0) = 2 - 3(0)^2 = 2 \text{ rad/s} \]

### Step 4: Find angular acceleration
Angular acceleration (\(\alpha\)) is the derivative of angular velocity with respect to time:
\[ \alpha(t) = \frac{d\omega}{dt} \]
Calculating the derivative:
\[ \alpha(t) = \frac{d}{dt}(2 - 3t^2) = -6t \]

### Step 5: Calculate angular acceleration at \(t = 1\)
Substituting \(t = 1\) into the angular acceleration equation:
\[ \alpha(1) = -6(1) = -6 \text{ rad/s}^2 \]

### Step 6: Find average angular acceleration during the first second
The average angular acceleration (\(\alpha_{\text{avg}}\)) over the time interval from \(t = 0\) to \(t = 1\) is given by:
\[ \alpha_{\text{avg}} = \frac{\Delta \omega}{\Delta t} \]
Where \(\Delta \omega = \omega(1) - \omega(0)\).

First, we need to calculate \(\omega(1)\):
\[ \omega(1) = 2 - 3(1)^2 = 2 - 3 = -1 \text{ rad/s} \]

Now, substituting the values:
\[ \Delta \omega = -1 - 2 = -3 \text{ rad/s} \]
\[ \Delta t = 1 - 0 = 1 \text{ s} \]
Thus,
\[ \alpha_{\text{avg}} = \frac{-3}{1} = -3 \text{ rad/s}^2 \]

### Final Result
The average angular acceleration during the first second is:
\[ \alpha_{\text{avg}} = -3 \text{ rad/s}^2 \]

---