A stationary wheel starts rotating about its own axis at uniform rate amgular acceleration `8rad//s^(2)`.The time taken by its to complete `77` rotation is

Options

A. `5.5 sec`
B. `7 sec`
C. `11 sec`
D. `14 sec`

Explanation

To solve the problem of finding the time taken by a stationary wheel to complete 77 rotations with a uniform angular acceleration of \(8 \, \text{rad/s}^2\), we can follow these steps:

### Step 1: Understand the given parameters
- The wheel starts from rest, so the initial angular velocity (\(\omega_i\)) is \(0 \, \text{rad/s}\).
- The angular acceleration (\(\alpha\)) is \(8 \, \text{rad/s}^2\).
- The total number of rotations (\(n\)) is \(77\).

### Step 2: Convert rotations to radians
Since we need to work in radians, we convert the number of rotations to radians:
\[ \theta = n \times 2\pi = 77 \times 2\pi = 154\pi \, \text{radians} \]

### Step 3: Use the angular displacement formula
The formula for angular displacement when starting from rest and under uniform angular acceleration is:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Substituting the known values:
\[ 154\pi = 0 \cdot t + \frac{1}{2} \cdot 8 \cdot t^2 \]
This simplifies to:
\[ 154\pi = 4t^2 \]

### Step 4: Solve for \(t^2\)
Rearranging the equation gives:
\[ t^2 = \frac{154\pi}{4} = 38.5\pi \]

### Step 5: Calculate \(t\)
Taking the square root of both sides:
\[ t = \sqrt{38.5\pi} \]

### Step 6: Substitute the value of \(\pi\)
Using \(\pi \approx 3.14\):
\[ t = \sqrt{38.5 \times 3.14} \approx \sqrt{120.59} \approx 10.98 \, \text{seconds} \]

### Step 7: Round to appropriate significant figures
Rounding to two decimal places, we find:
\[ t \approx 11 \, \text{seconds} \]

### Final Answer
The time taken by the wheel to complete 77 rotations is approximately **11 seconds**.

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