The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the b

Q: Why doesn't the decreasing mass of the water decrease the time period?

The time period of a simple pendulum is independent of its mass ($T = 2\pi\sqrt{l/g}$). It only depends on the effective length and gravity.

Q: Why does the center of gravity move down?

As water drains, the top part of the sphere becomes empty while the bottom part still has water. This uneven distribution of mass pulls the overall center of gravity downwards.

Q: At what exact point is the time period maximum?

The time period is maximum when the center of gravity is at its lowest point. This happens when the CG of the remaining water exactly coincides with the CG of the entire system.

Q: What happens when the sphere is completely empty?

When completely empty, the object is just a uniform hollow sphere. The center of gravity of a uniform hollow sphere is exactly at its geometric center, just like when it was completely full.

Q: Would the result change if the liquid was denser, like mercury?

No, the qualitative behavior would be exactly the same. The time period would still first increase and then decrease back to the original value, although the maximum time period reached might be different.

Question

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

Options:

(A) first increase and then decrease to the original value

(B) first decrease and then increase to the original value

(C) remain unchanged

(D) increase towards a saturation value

Sigma Percentile · Accepted Answer

### The Illusion of Mass

Imagine you are swinging on a swing. Does it matter if you are holding a heavy backpack? Surprisingly, no! The time period of a simple pendulum is famously independent of its mass.

The master equation governing this motion is:

$$T = 2\pi \sqrt{\frac{l}{g}}$$

Here, $g$ is the acceleration due to gravity, and $l$ is the **effective length** of the pendulum. But here is the catch: $l$ is not just the length of the string. It is the exact distance from the pivot point to the **Center of Gravity (CG)** of the bob.

### The Full Sphere

When our spherical hollow ball is completely filled with water, the mass is distributed perfectly symmetrically.

Because of this uniform distribution, the center of gravity lies exactly at the geometric center of the sphere. Let's call the distance from the pivot to this center $l$. The initial time period is simply based on this length $l$.

### The Great Drain

Now, we pull the plug. Water starts draining from the bottom. What happens to the mass distribution?

The top half of the sphere starts emptying out, while the bottom half still contains water. The system is no longer symmetric. The bottom is now significantly heavier than the top.

Because the mass is concentrated lower down, the overall **Center of Gravity shifts downwards**. The new effective length becomes $l + \Delta l$.

Looking back at our master equation, since the effective length has increased, the time period $T$ must also **increase**. The pendulum starts swinging slower!

### The Empty Shell

But the water doesn't drain forever. Eventually, the last drop falls out. What are we left with?

We are left with a perfectly uniform, empty hollow shell. And where is the center of gravity of a uniform hollow shell? It is right back at its geometric center!

The center of gravity, which had wandered downwards, is forced to travel back up to its original position. The effective length decreases back to $l$.

Consequently, the time period decreases and returns exactly to its original value.

### The Final Verdict

This creates a beautiful, continuous physical journey. As the water drains, the center of gravity dips down and then rises back up.

Therefore, the time period **first increases**, reaches a maximum when the CG is at its lowest point, and then **decreases back to the original value** once the sphere is completely empty.

Animated Solution for Physics - Oscillations: The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

Select Answer:

Visualized Solution

$T = 2 π \frac{l _{e f f}}{g}$

CG Shifts Downwards

$l_{e f f}$ Increases $\Rightarrow$ $T$ Increases

CG Returns to Center

$l_{e f f}$ Decreases $\Rightarrow$ $T$ Decreases

Final Conclusion

The Sigma Insight: Simple Harmonic Motion (SHM)

The Illusion of Mass

The Full Sphere

The Great Drain

The Empty Shell

The Final Verdict

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