Animated Solution for Physics - Oscillations: A wooden cube (density of wood $d$) of side $l$ floats in a liquid of density $\rho$ with its upper and lower surfaces horizontal. If the cube is pushed slightly down and released, it performs simple harmonic motion of period, $T$. Then, $T$ is equal to

Imagine a serene pond, perfectly still. Now, picture a wooden cube floating peacefully on its surface. It sits there in perfect equilibrium, a silent truce between gravity pulling it down and the water pushing it up. But what happens if we disrupt this peace? What if we give the cube a gentle nudge downwards?

Suddenly, the cube springs to life, bobbing up and down in a rhythmic dance. This isn't just random splashing; it's a beautiful, mathematically precise phenomenon known as Simple Harmonic Motion (SHM). Let's dive into the physics of this aquatic dance and uncover the hidden 'spring' that makes it all possible.

Equilibrium

Before we disturb the cube, let's understand its resting state. The cube has a side length of $l$ and is made of wood with density $d$.

We know that mass is simply volume multiplied by density. Therefore, the mass of our wooden cube is:

In this state of equilibrium, the downward weight of the cube ($mg$) is perfectly balanced by the upward buoyant force exerted by the liquid. The net force is zero, and the cube is happy.

Disturbing the Peace

Now, let's introduce a disturbance. We gently push the cube downwards by a small distance, $x$.

As the cube goes deeper, it displaces an additional volume of liquid. Think about the shape of this extra submerged part: it's a thin rectangular slice with a base area of $l^2$ and a height of $x$.

Therefore, the extra volume of liquid displaced is:

The Restoring Force

Here is where Archimedes' principle takes the spotlight. The principle states that the buoyant force on an object is equal to the weight of the fluid it displaces. Because we pushed the cube deeper, it displaced extra liquid, which means the liquid fights back with an extra upward buoyant force!

This extra buoyant force acts as our restoring force, desperately trying to push the cube back to its equilibrium position. Let's calculate it. The weight of the extra displaced liquid is its volume multiplied by the liquid's density ($\rho$) and gravity ($g$).

Why the negative sign? It's the soul of oscillations! It signifies that the force is always directed opposite to the displacement. If you push the cube down (negative displacement), the force pushes it up (positive direction).

Calculating the Time Period

Have you ever tried pushing a beach ball underwater? You feel a strong resistance pushing back against your hands. The deeper you push, the harder it fights back. This perfectly linear restoring force turns the entire ocean into a giant, invisible spring!

In physics, the hallmark of a spring-like system (Simple Harmonic Motion) is the equation $F=-kx$, where $k$ is the 'stiffness' or spring constant. By comparing our restoring force equation with this standard form, we can find the effective spring constant of our 'water spring':

Now, we hold the key to the rhythm. The time period $T$ of any Simple Harmonic Oscillator is given by the famous formula:

Let's substitute our expressions for mass $m$ and the effective spring constant $k$:

With a satisfying algebraic cancellation, the $l^2$ in the denominator wipes out two powers of $l$ in the numerator, leaving us with a beautifully elegant final result:

And there we have it! The time period of the bobbing cube depends on its size and the ratio of the densities, revealing the hidden harmony of floating objects.