Q.A piece of wood of mass $m$ is floating erect in a liquid whose density is $\rho$. If it is slightly pressed down and released, then it executes simple harmonic motion. Show that its time period of oscillation is $T = 2\pi\sqrt{\dfrac{m}{A\rho g}}$

Archimedes' Principle

Push a beach ball underwater in a swimming pool and feel it fight back — the deeper it goes, the harder it pushes up. Or try lifting a heavy stone while it's submerged: it feels far lighter than it does on land. Something about being surrounded by water makes objects seem to lose weight. That "something" is Archimedes' Principle.

The Intuition

A fluid is made of molecules pressing on everything around them, and pressure increases with depth. So the fluid beneath a submerged object pushes up on it harder than the fluid above pushes down — because there is more fluid weighing down from above onto the fluid below. This net upward push is the buoyant force.

The key insight: this upward force doesn't depend on the object's shape or material. It depends only on how much fluid the object displaces — the volume of fluid it pushes out of the way to make room for itself.

The Precise Statement

where $F_b$ is the buoyant force, $\rho_{\text{fluid}}$ is the fluid's density, $V_{\text{submerged}}$ is the submerged volume of the object, and $g$ is the acceleration due to gravity.

Archimedes' Principle: any object, wholly or partly immersed in a fluid, experiences an upward buoyant force equal to the weight of the fluid it displaces.

Why This Matters

What Decides Sink or Float: Density

• Object density greater than fluid density $\Rightarrow$ it sinks (weight exceeds buoyant force).
• Object density less than fluid density $\Rightarrow$ it floats (weight is less than buoyant force).
• Object density equal to fluid density $\Rightarrow$ it hovers at any depth. …