A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration At one instant it is rotating at 12 rad/s and after 80 radian of more angular displacement. its angular speed becomes 28 rad/s. How much time (seconds) does the disk takes to complete the mentioned angular displacement of 80 radians.

Answer

To solve the problem step by step, we will use the equations of motion for rotational motion.

### Step 1: Identify the given values
- Initial angular velocity, \( \omega_0 = 12 \, \text{rad/s} \)
- Final angular velocity, \( \omega = 28 \, \text{rad/s} \)
- Angular displacement, \( \theta = 80 \, \text{radians} \)

### Step 2: Use the angular motion equation
We will use the following equation of motion for angular displacement:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
where \( \alpha \) is the angular acceleration.

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ (28)^2 = (12)^2 + 2\alpha(80) \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 784 = 144 + 160\alpha \]

### Step 5: Rearrange the equation to solve for \( \alpha \)
Rearranging gives:
\[ 784 - 144 = 160\alpha \]
\[ 640 = 160\alpha \]

### Step 6: Solve for \( \alpha \)
Dividing both sides by 160:
\[ \alpha = \frac{640}{160} = 4 \, \text{rad/s}^2 \]

### Step 7: Use the angular velocity equation to find time
Now, we will use the equation:
\[ \omega = \omega_0 + \alpha t \]
Substituting the known values:
\[ 28 = 12 + 4t \]

### Step 8: Rearrange to solve for \( t \)
Rearranging gives:
\[ 28 - 12 = 4t \]
\[ 16 = 4t \]

### Step 9: Solve for \( t \)
Dividing both sides by 4:
\[ t = \frac{16}{4} = 4 \, \text{seconds} \]

### Final Answer
The disk takes **4 seconds** to complete the mentioned angular displacement of 80 radians.

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