Archimedes' Principle
Archimedes' Principle states that any object fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. This principle was discovered by Archimedes of Syracuse around 246 BCE, reportedly while stepping into a bath and observing water overflow. It is fundamental to naval architecture, submarine design, hot-air balloon operation, and density measurement techniques.
Key Formula
Fb = Weight of fluid displaced = ρ_fluid × V_submerged × g
LaTeX: F_b = W_{\text{fluid displaced}} = \rho_f V_s g
| Symbol | Meaning | Unit |
|---|---|---|
| F_b | Buoyant (upthrust) force | N |
| \rho_f | Density of the fluid | kg/m³ |
| V_s | Volume of fluid displaced | m³ |
| g | Gravitational acceleration | m/s² |
Worked Example
Problem
A gold crown (density 19,320 kg/m³) with a mass of 0.500 kg is weighed in air and then in water (ρ = 1000 kg/m³). What is its apparent weight in water, and would it pass the test for pure gold if its apparent weight in water is measured as 4.64 N?
Solution
Step 1: True weight = mg = 0.500 × 9.8 = 4.90 N. Step 2: Volume of crown = m/ρ_gold = 0.500/19320 = 2.59 × 10⁻⁵ m³. Step 3: Buoyant force = ρ_water × V × g = 1000 × 2.59×10⁻⁵ × 9.8 = 0.254 N. Step 4: Expected apparent weight = 4.90 − 0.254 = 4.646 N. Step 5: Measured apparent weight 4.64 N ≈ expected; crown is likely pure gold.
Answer
Apparent weight in water ≈ 4.65 N. The crown passes the Archimedes test for gold.
Verification of Archimedes' Principle for objects of different densities in water
| Object | Density (kg/m³) | Volume (cm³) | Weight (N) | Buoyant Force (N) | Behavior |
|---|---|---|---|---|---|
| Cork | 200 | 100 | 0.196 | 0.981 | Floats (20% submerged) |
| Wood (oak) | 720 | 100 | 0.706 | 0.981 | Floats (72% submerged) |
| Ice | 917 | 100 | 0.899 | 0.981 | Floats (93% submerged) |
| Aluminium | 2700 | 100 | 2.648 | 0.981 | Sinks |
| Iron | 7870 | 100 | 7.714 | 0.981 | Sinks rapidly |
| Gold | 19,320 | 100 | 18.95 | 0.981 | Sinks rapidly |
Interactive Tools
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Related Terms
Buoyancy
Buoyancy is the upward force exerted by a fluid on an object submerged or partially submerged in it, opposing the weight of the object. This force arises because fluid pressure increases with depth, so the pressure on the bottom face of a submerged object is greater than on the top face, creating a net upward force. An object floats when the buoyant force equals its weight, and sinks when its weight exceeds the buoyant force.
Pascal's Law
Pascal's Law states that pressure applied to an enclosed fluid is transmitted equally and undiminished in all directions throughout the fluid and acts with equal force per unit area on all surfaces of the container. Formulated by Blaise Pascal in 1653, this principle is the foundation of hydraulic systems used in hydraulic lifts, car brakes, excavator arms, and hydraulic jacks. The law assumes the fluid is incompressible and at rest (hydrostatic conditions).
Fluid Pressure
Fluid pressure is the force exerted per unit area by a fluid on any surface in contact with it, arising from the continuous collisions of fluid molecules. In a static fluid, pressure at a given depth depends on the fluid's density, gravitational acceleration, and the depth below the free surface. It is fundamental to hydraulics, hydrostatics, and the design of dams, pipelines, and pressure vessels.
Named after the Greek mathematician and inventor Archimedes of Syracuse (c. 287–212 BCE). The word comes from the Greek name 'Arkhimedes,' combining 'arkhi-' (master) and 'medes' (to think/plan). According to legend, Archimedes shouted 'Eureka!' (I have found it!) upon making this discovery.