The period of small oscillations of a simple pendulum of length `l` if its point of suspension `O` moves `a` with a constant acceleration `alpha = alpha_(1)overset(wedge)(i) - alpha_(2)overset(wedge)(j)` with respect to earth is (`overset(wedge)(i)` and `overset(wedge)(j)` are unit vectors in horizontal in horizontal and vertically upward directions respectively)

Options

A. `T=2pisqrt((l)/({(g-alpha_(2))^(2) + alpha_(1)^(2)}^(1//2)))`
B. `T=2pisqrt((l)/({(g-alpha_(1))^(2) + alpha_(2)^(2)}^(1//2)))`
C. `T=2pisqrt((l)/(g))`
D. `T=2pisqrt((l)/((g^(2) + alpha_(1)^(2))^(1//2)))`

Explanation

To find the period of small oscillations of a simple pendulum whose point of suspension is moving with constant acceleration, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Forces Acting on the Pendulum**:
The pendulum is affected by gravitational force acting downward (denoted as \( g \)) and the acceleration of the point of suspension \( O \) which is moving with acceleration \( \alpha = \alpha_1 \hat{i} - \alpha_2 \hat{j} \).

2. **Determine the Effective Acceleration**:
The effective acceleration acting on the pendulum bob can be determined by considering both the gravitational acceleration and the acceleration of the point of suspension. The net acceleration \( \vec{g}_{\text{effective}} \) can be expressed as:
\[ \vec{g}_{\text{effective}} = -\alpha_1 \hat{i} + (g - \alpha_2) \hat{j} \]
Here, \( -\alpha_1 \hat{i} \) represents the horizontal component and \( (g - \alpha_2) \hat{j} \) represents the vertical component.

3. **Calculate the Magnitude of the Effective Acceleration**:
The magnitude of the effective acceleration \( g_{\text{effective}} \) is given by:
\[ g_{\text{effective}} = \sqrt{(-\alpha_1)^2 + (g - \alpha_2)^2} \]
Simplifying this gives:
\[ g_{\text{effective}} = \sqrt{\alpha_1^2 + (g - \alpha_2)^2} \]

4. **Use the Formula for the Period of a Pendulum**:
The period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} \]
Substituting our expression for \( g_{\text{effective}} \) into this formula, we have:
\[ T = 2\pi \sqrt{\frac{L}{\sqrt{\alpha_1^2 + (g - \alpha_2)^2}}} \]

5. **Final Expression for the Period**:
To simplify, we can express the period as:
\[ T = 2\pi \sqrt{\frac{L}{\sqrt{\alpha_1^2 + (g - \alpha_2)^2}}} \]

### Conclusion
The period of small oscillations of the pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{\sqrt{\alpha_1^2 + (g - \alpha_2)^2}}} \]