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

Equilibrium of Forces

Upthrust & Archimedes’ Principle

🏊‍♂️ Imagine you’re in a swimming pool and you hold a small rock. You feel the water pushing you up – that’s upthrust. Archimedes discovered that the upward force on any object submerged in a fluid equals the weight of the fluid it displaces.



Mathematically, the upthrust (or buoyant force) is given by:

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

  • \$\rho\$ – density of the fluid (kg m⁻³)

  • \$g\$ – acceleration due to gravity (≈9.81 m s⁻²)

  • \$V\$ – volume of fluid displaced by the object (m³)

If the upthrust equals the weight of the object, the object floats. If it’s less, the object sinks; if more, it rises. This balance is the essence of equilibrium of forces.

Practical Example

🔍 Problem: A wooden block has a volume of 0.02 m³ and weighs 0.15 N. It is fully submerged in water (density ρ = 1000 kg m⁻³). What is the upthrust, and will the block float or sink?

  1. Calculate upthrust:

    \$F_{\text{up}} = 1000 \times 9.81 \times 0.02 = 196.2 \text{ N}\$

  2. Compare with weight:

    Weight = 0.15 N  (<< 196.2 N)

  3. Conclusion: Upthrust >> weight, so the block will rise and float (or even be pushed out of the water).

Fluid Density Table

FluidDensity (kg m⁻³)
Water (fresh)1000
Saltwater1025
Oil (vegetable)920
Mercury13546

Exam Tip:

• Always check units – density in kg m⁻³, volume in m³, g in m s⁻².

• Remember that the upthrust acts upward, opposing weight.

• For floating objects, set upthrust = weight to solve for unknowns.

• Use the symbol \$F_{\text{up}}\$ to avoid confusion with other forces.

• Practice quick mental calculations: ρ g ≈ 9.81 × ρ (kg m⁻³ s⁻²).