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, let's analyze the motion of the disc with uniform angular acceleration.

### Step 1: Understanding the given information
- The disc starts from rest, which means the initial angular velocity \( \omega_0 = 0 \).
- The disc rotates with uniform angular acceleration \( \alpha \).
- In the first 2 seconds, the angle rotated is \( \theta \).

### Step 2: Using the angular displacement formula
The formula for angular displacement \( \theta \) when starting from rest is given by:

\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Since \( \omega_0 = 0 \), the equation simplifies to:

\[ \theta = \frac{1}{2} \alpha t^2 \]

Substituting \( t = 2 \) seconds:

\[ \theta = \frac{1}{2} \alpha (2^2) = \frac{1}{2} \alpha (4) = 2\alpha \]

From this, we can express \( \alpha \):

\[ \alpha = \frac{\theta}{2} \]

### Step 3: Finding the angle rotated in the next 2 seconds
Now, we need to find the angle rotated in the next 2 seconds, which means we will calculate the angle rotated from \( t = 2 \) seconds to \( t = 4 \) seconds.

Using the same angular displacement formula for \( t = 4 \) seconds:

\[ \theta' = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Substituting \( \omega_0 = 0 \) and \( t = 4 \):

\[ \theta' = \frac{1}{2} \alpha (4^2) = \frac{1}{2} \alpha (16) = 8\alpha \]

### Step 4: Calculate the angle rotated in the first 4 seconds
Now, we know that the total angle rotated in the first 4 seconds is:

\[ \theta' = 8\alpha \]

### Step 5: Calculate the angle rotated in the first 2 seconds
From the previous calculation, the angle rotated in the first 2 seconds is:

\[ \theta = 2\alpha \]

### Step 6: Find the angle rotated in the next 2 seconds
The angle rotated in the next 2 seconds (from \( t = 2 \) to \( t = 4 \)) is given by:

\[ \text{Angle in next 2 seconds} = \theta' - \theta = 8\alpha - 2\alpha = 6\alpha \]

### Step 7: Substitute \( \alpha \)
Now substituting \( \alpha = \frac{\theta}{2} \):

\[ \text{Angle in next 2 seconds} = 6\left(\frac{\theta}{2}\right) = 3\theta \]

### Final Answer
Thus, the angle rotated by the disc in the next 2 seconds is:

\[ \text{Angle in next 2 seconds} = 3\theta \]