• $\rho_{\text{fluid}}$ = density of the fluid (not the object!)
• $V_{\text{disp}}$ = volume of fluid displaced by the object

💡 Key Idea: Buoyant force equals the weight of the fluid displaced.

Floating, Sinking, Suspended

For an object of density $\rho_{\text{obj}}$ in a fluid of density $\rho_{\text{fluid}}$:

For a floating object the buoyant force equals its weight:

$\rho_{\text{fluid}} g V_{\text{sub}} = \rho_{\text{obj}} g V_{\text{obj}}$

So the submerged fraction equals the density ratio:

$\frac{V_{\text{sub}}}{V_{\text{obj}}} = \frac{\rho_{\text{obj}}}{\rho_{\text{fluid}}}$

Apparent Weight

When an object is fully submerged but not floating freely:

$W_{\text{apparent}} = W_{\text{actual}} - F_b = (\rho_{\text{obj}} - \rho_{\text{fluid}}) g V$

This is what a scale reads when an object hangs from it underwater.

⚠️ Common Mistakes

1. Using the wrong density. Buoyancy uses the fluid's density, not the object's.
2. Forgetting partial submersion. A floating object only displaces enough fluid to balance its weight.
3. Confusing weight and apparent weight. True weight does not change underwater — only the net downward force does.

Key Formulas

Buoyant force (fully submerged → $V_{\text{disp}} = V$): $F_b = \rho_{\text{water}} g V = 1000(9.8)(1.85\times10^{-3}) \approx 18.1\text{ N}.$