A cabin is moving in a gravity free space along y-axis with an acceleration a. What is the time period of oscillation of a particle of mass m attached with an inextensible string of length l, in this cabin

Options

A. `2pi sqrt((l)/(g))`
B. `2pi sqrt((l)/(a))`
C. `2pi sqrt((l)/(a + g))`
D. `2pi sqrt((l)/(g-a))`

Explanation

To find the time period of oscillation of a particle of mass \( m \) attached to an inextensible string of length \( l \) in a cabin moving in a gravity-free space with an acceleration \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Environment**:
- The cabin is moving in a gravity-free space, which means the acceleration due to gravity \( g = 0 \).

2. **Identify the Forces**:
- In the cabin, the only force acting on the mass \( m \) is the tension \( T \) in the string. Since the cabin is accelerating upwards with acceleration \( a \), we can analyze the forces acting on the mass.

3. **Apply Newton's Second Law**:
- For the mass \( m \) in the cabin, the effective force acting on it can be considered as \( ma \) (due to the acceleration of the cabin). The tension in the string will provide the centripetal force required for the oscillation.

4. **Relate Tension to Acceleration**:
- The tension \( T \) in the string can be expressed as:
\[ T = ma \]

5. **Determine the Time Period of Oscillation**:
- The time period \( T \) of a simple harmonic oscillator is given by the formula:
\[ T = 2\pi \sqrt{\frac{l}{g_{\text{eff}}}} \]
- In this case, since we are in a gravity-free environment, we can replace \( g_{\text{eff}} \) with the acceleration \( a \) of the cabin:
\[ T = 2\pi \sqrt{\frac{l}{a}} \]

6. **Final Expression**:
- Thus, the time period of oscillation of the particle is:
\[ T = 2\pi \sqrt{\frac{l}{a}} \]

### Conclusion:
The time period of oscillation of the particle of mass \( m \) attached to the string of length \( l \) in the accelerating cabin is given by:
\[ T = 2\pi \sqrt{\frac{l}{a}} \]