Upthrust

Archimedes' Principle: This principle states that the upthrust acting on an object is equal to the weight of the fluid that the object displaces. It provides the mathematical link between volume and force.

Pressure-Depth Relationship: In a fluid at rest, pressure increases with depth according to $P = \rho gh$. Because the bottom surface of a submerged object is deeper than the top, it experiences a greater force ($F = P \times A$).

Displacement: When an object is immersed, it pushes aside a volume of fluid equal to the volume of the part of the object that is submerged. The mass of this displaced fluid is $m = \rho V$.

Calculating Upthrust: Use the formula $U = \rho V g$, where $\rho$ is the density of the fluid (not the object), $V$ is the volume of the displaced fluid, and $g$ is the gravitational field strength.

Determining Apparent Weight: When an object is in a fluid, its weight appears to decrease. The apparent weight is calculated as $W_{apparent} = W_{actual} - U$.

Equilibrium Analysis: For a floating object, the upthrust must exactly equal the object's weight ($U = W$). This allows for the calculation of the submerged volume fraction using the ratio of densities: $\frac{V_{submerged}}{V_{total}} = \frac{\rho_{object}}{\rho_{fluid}}$.

Object Density vs. Fluid Density: Upthrust depends solely on the fluid's density and the volume displaced. The object's density only determines if the weight will overcome that upthrust.

Total Volume vs. Submerged Volume: For fully submerged objects, $V$ is the object's total volume. For floating objects, $V$ is only the volume of the part below the fluid line.

Unit Consistency: Always ensure density is in $kg/m^3$ and volume is in $m^3$ before calculating force in Newtons. A common mistake is mixing $g/cm^3$ with meters.

Identify the Fluid: Ensure you use the density of the fluid ($ho_{fluid}$) in the upthrust formula, not the density of the object. Examiners often provide both to test this distinction.

Submerged Volume Check: If an object is floating, do not use its total volume to calculate upthrust. Use only the volume of the fluid it has displaced.

Sanity Check: If an object is described as 'floating', the upthrust must equal the weight. You can often solve these problems by equating $mg = \rho V g$ without needing the value of $g$.