A wheel which is initially at rest is subjected to a constant angular acceleration about its axis. It rotates through an angle of `15^@` in time `t` sec. Then how much it rotates in the next `2t` sec

Options

A. `90^(@)`
B. `120^(@)`
C. `30^(@)`
D. `45^(@)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a wheel that starts from rest and undergoes a constant angular acceleration. It rotates through an angle of \(15^\circ\) in time \(t\) seconds. We need to find out how much it rotates in the next \(2t\) seconds.

### Step 2: Convert the angle to radians
First, we convert the angle from degrees to radians because the standard unit for angular measurements in physics is radians.
\[ \theta = 15^\circ = 15 \times \frac{\pi}{180} = \frac{\pi}{12} \text{ radians} \]

### Step 3: Use the angular displacement formula
The angular displacement \(\theta\) under constant angular acceleration is given by the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Since the wheel is initially at rest, \(\omega_0 = 0\). Thus, the equation simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]
Substituting the angle we found:
\[ \frac{\pi}{12} = \frac{1}{2} \alpha t^2 \]

### Step 4: Solve for angular acceleration \(\alpha\)
Rearranging the equation to solve for \(\alpha\):
\[ \alpha = \frac{2 \theta}{t^2} = \frac{2 \times \frac{\pi}{12}}{t^2} = \frac{\pi}{6 t^2} \]

### Step 5: Calculate the angle rotated in the next \(2t\) seconds
Now, we need to find the total angle rotated in \(3t\) seconds and then subtract the angle rotated in the first \(t\) seconds.

Using the same formula for \(3t\):
\[ \theta_{3t} = \omega_0 (3t) + \frac{1}{2} \alpha (3t)^2 \]
Again, since \(\omega_0 = 0\):
\[ \theta_{3t} = \frac{1}{2} \alpha (3t)^2 = \frac{1}{2} \alpha (9t^2) = \frac{9}{2} \alpha t^2 \]
Substituting \(\alpha\):
\[ \theta_{3t} = \frac{9}{2} \left(\frac{\pi}{6 t^2}\right) t^2 = \frac{9\pi}{12} = \frac{3\pi}{4} \text{ radians} \]

### Step 6: Find the angle rotated in the next \(2t\) seconds
Now, we subtract the angle rotated in the first \(t\) seconds:
\[ \theta_{next\ 2t} = \theta_{3t} - \theta_{t} \]
\[ \theta_{next\ 2t} = \frac{3\pi}{4} - \frac{\pi}{12} \]
To subtract these, we need a common denominator:
\[ \frac{3\pi}{4} = \frac{9\pi}{12} \]
Thus:
\[ \theta_{next\ 2t} = \frac{9\pi}{12} - \frac{\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3} \text{ radians} \]

### Step 7: Convert the result back to degrees
Now, we convert \(\frac{2\pi}{3}\) radians back to degrees:
\[ \theta_{next\ 2t} = \frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ \]

### Final Answer
The wheel rotates \(120^\circ\) in the next \(2t\) seconds.

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