STATEMENT-1`:` Time periodof oscillation of a simple pendulum mounted in a cabin that is freely falling is zero
and
STATEMENT -2 `,` In the cabin falling freely under gravity the pendulum is in state of weightlessness.

Answer

To analyze the statements provided in the question, we will evaluate each statement step by step.

### Step 1: Understanding Statement 1
**Statement 1:** The time period of oscillation of a simple pendulum mounted in a cabin that is freely falling is zero.

- The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 2: Analyzing the Effect of Free Fall
- In a freely falling cabin, both the pendulum and the cabin are accelerating downwards at the same rate due to gravity. This means that the effective gravitational acceleration \( g' \) experienced by the pendulum is zero because the pendulum is in a state of free fall along with the cabin.

### Step 3: Substituting \( g' \) into the Formula
- If \( g' = 0 \), substituting this into the time period formula gives:
\[ T = 2\pi \sqrt{\frac{L}{0}} \]
This expression tends to infinity, indicating that the pendulum does not oscillate in a freely falling frame.

### Step 4: Conclusion for Statement 1
- Therefore, **Statement 1 is false** because the time period of oscillation is not zero; it is infinite.

### Step 5: Understanding Statement 2
**Statement 2:** In the cabin falling freely under gravity, the pendulum is in a state of weightlessness.

- Weightlessness occurs when the effective gravitational force acting on an object is zero. Since the pendulum is in a freely falling cabin, the gravitational force acting on it is countered by the acceleration of the cabin.

### Step 6: Conclusion for Statement 2
- As a result, the weight of the pendulum \( W = mg \) becomes zero in this scenario, confirming that the pendulum is indeed in a state of weightlessness. Thus, **Statement 2 is true**.

### Final Conclusion
- **Statement 1 is false** and **Statement 2 is true**.