Two wheels, each marked with a dot on its rim, are mounted side by side. Initially the dots are alinged and wheels are at rest. One of the wheels is given a constant angular acceleration of `(pi//2) rad//sec^(2)` and the other wheel is given a constant angular acceleration `(pi//4)rad//sec^(2)`. Both acceleration are in the same direction. Find the time (in s) after which the two dots will becomes alingned again for the first time.

Explanation

To solve the problem of when the two dots on the wheels will align again, we can follow these steps:

### Step 1: Understand the problem
We have two wheels with different angular accelerations. We need to find the time at which the dots on their rims will align again after being initially aligned.

### Step 2: Define the angular displacements
Let:
- \( \alpha_1 = \frac{\pi}{2} \, \text{rad/s}^2 \) (angular acceleration of the first wheel)
- \( \alpha_2 = \frac{\pi}{4} \, \text{rad/s}^2 \) (angular acceleration of the second wheel)

The angular displacement \( \theta \) for each wheel can be calculated using the formula:
\[ \theta = \frac{1}{2} \alpha t^2 \]

### Step 3: Write the equations for angular displacements
For the first wheel:
\[ \theta_1 = \frac{1}{2} \alpha_1 t^2 = \frac{1}{2} \cdot \frac{\pi}{2} \cdot t^2 = \frac{\pi}{4} t^2 \]

For the second wheel:
\[ \theta_2 = \frac{1}{2} \alpha_2 t^2 = \frac{1}{2} \cdot \frac{\pi}{4} \cdot t^2 = \frac{\pi}{8} t^2 \]

### Step 4: Set the condition for alignment
The two dots will align again when the angular displacements are equal to an integer multiple of \( 2\pi \):
\[ \theta_1 = 2\pi n_1 \quad \text{and} \quad \theta_2 = 2\pi n_2 \]
where \( n_1 \) and \( n_2 \) are integers representing the number of complete rotations.

### Step 5: Relate the two equations
From the equations for \( \theta_1 \) and \( \theta_2 \):
\[ \frac{\pi}{4} t^2 = 2\pi n_1 \quad \text{(1)} \]
\[ \frac{\pi}{8} t^2 = 2\pi n_2 \quad \text{(2)} \]

### Step 6: Simplify the equations
From equation (1):
\[ t^2 = 8 n_1 \]
From equation (2):
\[ t^2 = 16 n_2 \]

### Step 7: Set the equations equal to each other
Since both expressions equal \( t^2 \):
\[ 8 n_1 = 16 n_2 \]
This simplifies to:
\[ \frac{n_1}{n_2} = \frac{16}{8} = 2 \]
Thus, we can set \( n_1 = 2 \) and \( n_2 = 1 \).

### Step 8: Substitute back to find \( t \)
Substituting \( n_1 = 2 \) into the equation for \( t^2 \):
\[ t^2 = 8 \cdot 2 = 16 \]
Thus,
\[ t = \sqrt{16} = 4 \, \text{s} \]

### Final Answer
The time after which the two dots will become aligned again for the first time is **4 seconds**.

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