A disc initially at rest , is rotated about its axis with uniform angular acceleration . In the first 2 s, it rotates an angle `theta` . In the next 2s, the disc rotates through an angle

Options

A. `theta`
B. `2theta`
C. `3theta`
D. `4theta`

Answer

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

### Step 1: Understand the problem
We have a disc that starts from rest and rotates with a uniform angular acceleration. We need to find the angle it rotates through in the second interval of 2 seconds, given that it rotates through an angle θ in the first 2 seconds.

### Step 2: Use the equation of motion for rotation
The angle θ rotated in time \( t \) under uniform angular acceleration is given by the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
where:
- \( \theta \) is the angle rotated,
- \( \omega_0 \) is the initial angular velocity,
- \( \alpha \) is the angular acceleration,
- \( t \) is the time.

### Step 3: Calculate the angle for the first 2 seconds
Since the disc starts from rest, \( \omega_0 = 0 \). For the first 2 seconds:
\[ \theta = 0 \cdot 2 + \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha \cdot 4 = 2\alpha \]
Thus, we have:
\[ \theta = 2\alpha \quad \text{(1)} \]

### Step 4: Find the angular acceleration
From equation (1), we can express angular acceleration \( \alpha \):
\[ \alpha = \frac{\theta}{2} \quad \text{(2)} \]

### Step 5: Calculate the angle for the next 2 seconds
Now we need to find the angle rotated in the next 2 seconds (from \( t = 2 \) s to \( t = 4 \) s). The total time at this point is 4 seconds. We can calculate the total angle rotated in 4 seconds using the same formula:
\[ \theta_{\text{total}} = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Substituting \( t = 4 \) seconds:
\[ \theta_{\text{total}} = 0 \cdot 4 + \frac{1}{2} \alpha (4^2) = \frac{1}{2} \alpha \cdot 16 = 8\alpha \]

### Step 6: Find the angle for the second interval
The angle rotated in the second interval (from 2s to 4s) is given by:
\[ \theta_{\text{next}} = \theta_{\text{total}} - \theta = 8\alpha - 2\alpha = 6\alpha \]

### Step 7: Substitute the value of \( \alpha \)
Now, substituting \( \alpha \) from equation (2):
\[ \theta_{\text{next}} = 6 \cdot \frac{\theta}{2} = 3\theta \]

### Conclusion
Thus, the angle rotated by the disc in the next 2 seconds is \( 3\theta \).

### Final Answer
The answer is \( 3\theta \).

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