The buoyant force on an object equals the weight of the fluid it displaces.

\displaystyle F_B = \rho_{fluid} V_{sub}\, g

\displaystyle \text{fraction submerged} = \frac{\rho_{obj}}{\rho_{fluid}}

\displaystyle SG = \frac{\rho_{obj}}{\rho_{water}}

Buoyancy exists because pressure is higher at the bottom of an object than at the top.
F_B > weight → rises and floats. F_B < weight → sinks (but still feels buoyancy). Equal → hovers.
Specific gravity below 1 floats in water; above 1 sinks. For a floating object, the fraction submerged equals SG (ice: 0.917 in fresh water).
Apparent weight in water = real weight − F_B. Weighing in air and in water gives density.
Shape matters: steel shaped as a hull displaces enough water to float.

Problem	Setup	Answer
Floating woman, 97.0% submerged (Ex. 11.9)	ρ = 0.970 × 1000	970 kg/m³
Coin: 8.630 g in air, 7.800 g in water (Ex. 11.10)	V = 0.830 cm³; ρ = 8.630 / 0.830	10.4 g/cm³ (silver)
1.00 × 10⁷ kg of steel as a solid block (Ex. 11.8)	V = 1.28 × 10³ m³; F_B = 1.3 × 10⁷ N vs. weight 9.8 × 10⁷ N	Sinks; as a hull displacing 1.00 × 10⁵ m³ it floats