Animated Solution for Physics - Oscillations: A simple pendulum oscillating in air has period $T$. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}$th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is

Unlocking Effective Gravity

Imagine you are standing in a physics lab, observing a simple pendulum swinging back and forth in the air. Its rhythm is steady, governed by a fundamental law of nature. But what happens when you take that exact same pendulum and submerge it completely in a tank of liquid? Does it swing faster? Slower? Let's dive into the fascinating mechanics of this classic JEE problem.

To understand the motion of any simple pendulum, we must start with its master equation. The time period $T$ of a simple pendulum is given by:

Here, $L$ is the length of the string, and $g_{\text{eff}}$ is the effective acceleration due to gravity.

When the pendulum is swinging in the air, the only significant vertical force acting on the bob is gravity pulling it straight down. Therefore, the effective gravity is simply $g$. Our initial time period is beautifully simple:

Now, let's plunge the pendulum into the liquid. The environment has changed, and so have the forces. The bob still experiences the downward pull of gravity ($mg$), but now, the liquid fights back. According to Archimedes' principle, the liquid exerts an upward buoyant force ($F_B$) on the bob.

To find the new time period, we need to determine the net downward force acting on the bob. Let's break down the masses and forces using density and volume. Let $V$ be the volume of the bob and $\rho$ be its density. The mass of the bob is simply $m=V\rho$.

The downward gravitational force is $mg=V\rho g$.

The upward buoyant force depends on the weight of the displaced liquid. Since the bob is completely submerged, it displaces a volume $V$ of liquid. The problem states that the density of the liquid is $\frac{1}{16}$th of the density of the bob. So, ${\rho }_{\text{liquid}}=\frac{\rho }{16}$.

The buoyant force is:

Now, let's calculate the net downward force ($F_{\text{net}}$):

Substituting our expressions:

By factoring out $V\rho g$, we get:

This net force is what drives the pendulum's oscillation in the liquid. We can express this net force in terms of an effective gravity, $g_{\text{eff}}$, such that $F_{\text{net}}=m{g}_{\text{eff}}$. Since $m=V\rho$, we have:

Canceling $V\rho$ from both sides reveals our new effective gravity:

We have successfully unlocked the effective gravity! The final step is to substitute this back into our master equation for the time period:

Let's rearrange the fraction inside the square root:

Notice that the term in the parentheses is exactly our initial time period, $T$. Furthermore, the square root of $16$ is a perfect $4$. Pulling that out, we arrive at our elegant final answer:

By understanding how buoyancy alters the effective gravity, we can effortlessly predict the behavior of the pendulum in any fluid. This is the power of breaking down complex physical situations into fundamental forces!