calculate the upthrust acting on an object in a fluid using the equation F = ρgV (Archimedes’ principle)

Equilibrium of Forces

Learning Objective

Calculate the upthrust (buoyant force) acting on an object immersed in a fluid using the equation

\$F_{\text{up}} = \rho g V\$

where \$\rho\$ is the fluid density, \$g\$ is the acceleration due to gravity and \$V\$ is the volume of fluid displaced.

Key Concepts

• Upthrust (Buoyant Force): The upward force exerted by a fluid on an object immersed in it.
• Archimedes’ Principle: An object submerged in a fluid experiences an upthrust equal to the weight of the fluid displaced.
• Equilibrium: When the net force on an object is zero, i.e. \$\sum \vec F = 0\$.

Derivation of the Upthrust Formula

1. Consider a small element of fluid at depth \$h\$ with cross‑sectional area \$A\$ and thickness \$dh\$.
2. The pressure at depth \$h\$ is \$p = p0 + \rho g h\$, where \$p0\$ is the atmospheric pressure.
3. The upward force on the bottom face of the element is \$p\,A\$, and the downward force on the top face is \$(p - \rho g\,dh)A\$.
4. The net upward force on the element is \$\rho g A\,dh\$, which is \$\rho g\$ times the volume \$dV = A\,dh\$.
5. Integrating over the entire submerged volume \$V\$ gives \$F_{\text{up}} = \rho g V\$.

Applying the Formula – Worked Example

Problem: A solid cube of side \$0.10\,\text{m}\$ is fully submerged in water (\$\rho_{\text{water}} = 1000\,\text{kg\,m}^{-3}\$). Calculate the upthrust acting on the cube.

The upthrust equals \$9.81\,\text{N}\$, which is the weight of the water displaced.

Common Mistakes to Avoid

• Using the volume of the object instead of the volume of fluid displaced when the object is only partially submerged.
• Neglecting the effect of atmospheric pressure; remember that Archimedes’ principle concerns the *difference* in pressure between top and bottom surfaces.
• Confusing density (\$\rho\$) with mass; ensure you use \$\rho\$ in \$\text{kg\,m}^{-3}\$.

Practice Questions

1. A solid sphere of radius \$0.05\,\text{m}\$ is suspended in oil (\$\rho_{\text{oil}} = 800\,\text{kg\,m}^{-3}\$). Find the upthrust.
2. A wooden block of density \$600\,\text{kg\,m}^{-3}\$ and volume \$2.5\times10^{-3}\,\text{m}^3\$ is placed in water. Determine the magnitude of the upthrust and state whether the block will float or sink.
3. Explain why a submarine can dive by taking in water and surface by expelling it, using the upthrust formula.

Summary

In equilibrium, the upthrust \$F{\text{up}}\$ balances the weight of the object (or a component of it). The simple relation \$F{\text{up}} = \rho g V\$ allows rapid calculation of the buoyant force once the displaced volume and fluid density are known.