The angular displacement of a particle performing circular motion is
`theta=(t^(4))/(60)-(t)/(4)`
where `theta` is radian and 't' is in seconds. Then the acceleration of a particle at the end of 10 s will be

Options

A. `10 rad //s^(2)`
B. `20 rad//s^(2)`
C. `30 rad//s^(2)`
D. `15 rad//s^(2)`

Explanation

To find the acceleration of a particle performing circular motion given the angular displacement, we will follow these steps:

### Step 1: Write down the angular displacement equation
The angular displacement is given as:
\[ \theta(t) = \frac{t^4}{60} - \frac{t}{4} \]

### Step 2: Differentiate to find angular velocity
To find the angular velocity, we differentiate the angular displacement with respect to time \( t \):
\[ \omega(t) = \frac{d\theta}{dt} = \frac{d}{dt}\left(\frac{t^4}{60} - \frac{t}{4}\right) \]
Using the power rule of differentiation:
\[ \omega(t) = \frac{4t^3}{60} - \frac{1}{4} = \frac{t^3}{15} - \frac{1}{4} \]

### Step 3: Differentiate to find angular acceleration
Next, we differentiate the angular velocity to find the angular acceleration:
\[ \alpha(t) = \frac{d\omega}{dt} = \frac{d}{dt}\left(\frac{t^3}{15} - \frac{1}{4}\right) \]
Again using the power rule:
\[ \alpha(t) = \frac{3t^2}{15} = \frac{t^2}{5} \]

### Step 4: Calculate angular acceleration at \( t = 10 \) seconds
Now, we substitute \( t = 10 \) seconds into the angular acceleration equation:
\[ \alpha(10) = \frac{10^2}{5} = \frac{100}{5} = 20 \text{ rad/s}^2 \]

### Conclusion
Thus, the angular acceleration of the particle at the end of 10 seconds is:
\[ \alpha = 20 \text{ rad/s}^2 \]

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