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

Options

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

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the given data
- The wheel starts from rest, so the initial angular velocity \( \omega_0 = 0 \).
- The angular acceleration \( \alpha = 8 \, \text{rad/s}^2 \).
- The number of rotations \( N = 77 \).

### Step 2: Convert rotations to radians
To find the angular displacement \( \theta \) in radians, we use the formula:
\[ \theta = N \times 2\pi \]
Substituting the value of \( N \):
\[ \theta = 77 \times 2\pi = 154\pi \, \text{radians} \]

### Step 3: Use the angular displacement equation
We will use the angular displacement formula from kinematics:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Since the wheel starts from rest, \( \omega_0 = 0 \), so the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

### Step 4: Substitute known values into the equation
Now substituting for \( \theta \) and \( \alpha \):
\[ 154\pi = \frac{1}{2} \times 8 \times t^2 \]
This simplifies to:
\[ 154\pi = 4t^2 \]

### Step 5: Solve for \( t^2 \)
Rearranging the equation gives:
\[ t^2 = \frac{154\pi}{4} \]
Calculating \( t^2 \):
\[ t^2 = 38.5\pi \]

### Step 6: Calculate \( t \)
Taking the square root to find \( t \):
\[ t = \sqrt{38.5\pi} \]
Using \( \pi \approx 3.14 \):
\[ t = \sqrt{38.5 \times 3.14} \approx \sqrt{120.59} \approx 10.97 \, \text{seconds} \]

### Step 7: Round the answer
Rounding \( t \) gives approximately:
\[ t \approx 11 \, \text{seconds} \]

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

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