Archimedes' Principle and Buoyancy

When an object is immersed in a fluid the surrounding pressure pushes more strongly on its lower surface than on its upper. The net upward force is the buoyant force and equals the weight of fluid displaced. NEET problems on floating bodies and apparent weight rely entirely on this rule.

Concept

Archimedes' principle: a body wholly or partially immersed in a fluid experiences an upward force equal to the weight of fluid displaced.

• If $\rho_{\text{body}} > \rho_{\text{fluid}}$: the body sinks.
• If $\rho_{\text{body}} = \rho_{\text{fluid}}$: it remains in equilibrium at any depth.
• If $\rho_{\text{body}} < \rho_{\text{fluid}}$: it floats with only a fraction submerged.

For a floating body the submerged volume fraction equals the density ratio.

Formula Derivation

Consider a body of volume $V_{\text{sub}}$ submerged in a fluid of density $\rho_{f}$. The net upward force is

$F_{b} = V_{\text{sub}}\,\rho_{f}\,g = \text{weight of displaced fluid}.$

For a floating body of total volume $V$ and density $\rho_{b}$,

$\rho_{b} V g = \rho_{f} V_{\text{sub}} g \implies \frac{V_{\text{sub}}}{V} = \frac{\rho_{b}}{\rho_{f}}.$

The apparent weight in fluid is

$W_{\text{app}} = W - F_{b} = V g (\rho_{b} - \rho_{f}).$

NEET-style Worked Example

A block of wood of density $0.8\ \text{g/cm}^{3}$ floats in water. What fraction of its volume is above the water surface?

$\frac{V_{\text{sub}}}{V} = \frac{\rho_{b}}{\rho_{f}} = \frac{0.8}{1.0} = 0.8.$

So $20\%$ of the wood is above water.

Common Confusions

• Buoyancy depends on the fluid's density, not on the body's shape or composition.
• For an iron block held by a string in water, tension $T = W - F_{b}$, not $W$.
• A body fully submerged but supported by the bottom of the container exerts a normal force equal to apparent weight, not zero.
• In accelerating fluids, replace $g$ by effective gravity $g_{\text{eff}}$.

Key Takeaways

• $F_{b} = V_{\text{sub}}\rho_{f} g$ (weight of displaced fluid).
• Floating fraction submerged = $\rho_{b}/\rho_{f}$.
• Apparent weight $= W(1 - \rho_{f}/\rho_{b})$ when fully submerged.
• Icebergs float with $\sim 92\%$ submerged ($\rho_{\text{ice}}/\rho_{\text{water}} = 0.917$).
• Buoyancy is independent of pressure depth — it equals fluid weight displaced.