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Step-by-Step Explanation
Step 1: Identify the Principle of the Simple 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 distance (often referred to as the effective length) from the pivot point to the center of mass of the bob, and $g$ is the acceleration due to gravity.
Step 2: Understand the Effect of Water Draining on the Center of Mass
Initially, the bob is filled with water. When the hole at the bottom is unplugged, water begins to flow out of the bob. As water flows out, the distribution of mass inside the bob changes. Specifically, the center of mass of the “bob + water” system shifts downward first because the mass of water is redistributing within the spherical shell.
Step 3: Relate the Shift in Center of Mass to the Time Period
1. While the center of mass moves downward, the effective length $L$ increases slightly (say to $L + \Delta L$). According to the pendulum formula $T = 2\pi \sqrt{\frac{L}{g}}$, if $L$ increases, $T$ also increases. Thus, the time period first becomes larger.
2. As more water continues to drain, the center of mass eventually moves back towards its original position (once the water is nearly depleted, the remaining system’s center of mass rises back to the initial level). Consequently, the effective length returns to approximately its original value $L$, causing the time period to decrease back to its initial value.
Step 4: Conclude the Overall Effect on Time Period
Hence, during the process of water draining out, the time period:
First increases (due to the downward shift of center of mass, effectively increasing $L$).
Then decreases back to the original value (as the center of mass returns to its initial position once the water has drained out).
Final Answer
The time period of the pendulum first increases and then decreases to the original value.
