To solve the problem, we need to analyze the assertion and reason provided in the question step by step.
### Step 1: Understanding the Assertion
The assertion states that if a fine hole is made in the bottom of a ball (which is the bob of a simple pendulum filled with water), the time period of the pendulum first increases and then decreases.
### Step 2: Analyzing the Time Period of a Pendulum
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 3: Initial Condition
Initially, the bob is full of water, and we denote its mass as \( M \). The center of mass of the bob is at its center, and the effective length of the pendulum is \( L \).
### Step 4: Effect of Water Flowing Out
When a fine hole is made in the bottom of the ball, water starts to flow out. As water flows out, the mass of the bob decreases, which affects the center of mass.
1. **First Phase (Water Flowing Out)**:
- As water flows out, the center of mass of the bob shifts downward because the remaining water is still at the bottom of the ball.
- This effectively increases the length of the pendulum to \( L + \Delta L \) (where \( \Delta L \) is the increase in length due to the shift in the center of mass).
- Consequently, the time period increases:
\[
T_{increased} = 2\pi \sqrt{\frac{L + \Delta L}{g}}
\]
2. **Second Phase (Less than Half Full)**:
- Once enough water has flowed out, the center of mass shifts upward as the amount of water decreases.
- Now, the effective length of the pendulum decreases to \( L - \Delta L \).
- The time period then decreases:
\[
T_{decreased} = 2\pi \sqrt{\frac{L - \Delta L}{g}}
\]
### Step 5: Conclusion
From the analysis, we conclude that:
- The time period first increases as water flows out and then decreases when the bob is less than half full.
- Therefore, the assertion is correct.
### Step 6: Analyzing the Reason
The reason states that as water flows out of the bob, the weight of the bob decreases. This is also true, as the mass of the bob decreases with the loss of water.
### Final Conclusion
Both the assertion and reason are correct, but the reason does not adequately explain the assertion. Therefore, the correct answer is that both statements are true, but the reason is not the correct explanation for the assertion.
