Archimedes' Principle

The Mathematical Formula for the buoyant force is derived from the weight of the displaced fluid: $F_b = \rho_{fluid} \cdot V_{submerged} \cdot g$. Here, $\rho_{fluid}$ is the density of the fluid, $V_{submerged}$ is the volume of the object below the fluid line, and $g$ is the acceleration due to gravity.

The Pressure Gradient is the physical cause of buoyancy. Since pressure $P = \rho gh$, the pressure at the bottom of a submerged object ($h_{bottom}$) is greater than at the top ($h_{top}$), resulting in a net upward force.

Apparent Weight is the weight an object seems to have when submerged in a fluid. It is calculated as the difference between the actual weight in a vacuum (or air) and the buoyant force: $W_{apparent} = W_{actual} - F_b$.

Determining Floatation: To predict if an object will float, compare the density of the object ($\rho_{obj}$) to the density of the fluid ($\rho_{fluid}$). If $\rho_{obj} < \rho_{fluid}$, the object will float; if $\rho_{obj} > \rho_{fluid}$, it will sink.

Calculating Submerged Volume: For a floating object, the fraction of the object's volume that is submerged is equal to the ratio of the densities: $\frac{V_{submerged}}{V_{total}} = \frac{\rho_{obj}}{\rho_{fluid}}$.

Measuring Density (Hydrostatic Weighing): By measuring an object's weight in air and its apparent weight in water, one can calculate its volume and density without complex geometric measurements.

Mass of Object vs. Mass of Displaced Fluid: For a floating object, these two masses are exactly equal. For a sinking object, the mass of the object is greater than the mass of the displaced fluid.

Total Volume vs. Submerged Volume: When calculating $F_b$, only the volume actually under the fluid surface matters. If an object is 50% submerged, only that 50% contributes to the buoyant force.

Check the Density: Always ensure you are using the density of the fluid when calculating the buoyant force ($F_b$). A common mistake is using the density of the submerged object instead.

Unit Consistency: Verify that density is in $kg/m^3$ and volume is in $m^3$ if you are using $g = 9.8 m/s^2$. If using $g/cm^3$, ensure volume is in $cm^3$.

Equilibrium Condition: For any object floating at rest, the buoyant force must exactly equal the object's weight ($F_b = W$). Use this equality to solve for unknown densities or volumes.

Sanity Check: If an object is less dense than the fluid, your calculated $W_{apparent}$ should be zero (it floats). If it is more dense, $W_{apparent}$ must be positive but less than $W_{actual}$.