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Archimedes' Principle Calculator

Archimedes' principle reveals why objects behave differently in fluids—some float effortlessly while others plummet. This calculator determines the buoyant force acting on submerged objects and calculates key properties like apparent weight, displaced fluid volume, and object density. Engineers use it to design vessels and submarines; geologists apply it to identify mineral purity; students rely on it to grasp fluid mechanics fundamentals.

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Creators Steven Wooding, BSc Physics
4,723 people find this calculator helpful

Understanding Archimedes' Principle

Archimedes' principle states that an object immersed in a fluid experiences an upward force equal to the weight of fluid it displaces. This happens because pressure increases with depth in a fluid. The pressure pushing upward on an object's bottom surface exceeds the pressure pushing downward on its top, creating a net upward force called the buoyant force.

The principle applies to all fluids—liquids and gases alike. A helium balloon rises through air, and a submarine hovers in water, both obeying the same fundamental physics. The magnitude of buoyancy depends entirely on three factors: how much fluid is displaced, the fluid's density, and gravitational acceleration.

Whether an object floats, sinks, or hovers depends on comparing its weight to the buoyant force. When weight exceeds buoyancy, the object sinks. When buoyancy wins, it floats. They balance for neutral buoyancy.

Buoyant Force and Apparent Weight

The buoyant force equals the weight of displaced fluid. When an object is submerged, its apparent weight (what a scale reads in fluid) drops below its true weight by exactly this buoyant force amount.

FB = Vfluid × g × ρfluid

Wapparent = Wtrue − FB

Wapparent = Vobject × g × (ρobject − ρfluid)

  • F<sub>B</sub> — Buoyant force in newtons
  • V<sub>fluid</sub> — Volume of fluid displaced in cubic metres
  • g — Acceleration due to gravity (9.8 m/s² on Earth)
  • ρ<sub>fluid</sub> — Density of the fluid in kg/m³
  • W<sub>true</sub> — True weight of object in air in newtons
  • W<sub>apparent</sub> — Apparent weight in fluid in newtons
  • ρ<sub>object</sub> — Density of the object in kg/m³

Practical Applications Across Industries

Naval architects exploit Archimedes' principle to design vessels that displace water equal to their weight, enabling massive steel ships to float. Submarines control buoyancy by adjusting ballast water volume, achieving neutral buoyancy at desired depths.

Geologists and mineralogists use hydrometers—devices that float at different depths in liquids—to determine fluid density and identify mineral composition. A piece of ore's density compared to water density reveals its purity and mineral concentration.

Hot air balloons rely entirely on this principle: heating air reduces its density below surrounding atmosphere density, creating buoyancy. Aircraft similarly generate lift by displacing air mass through wing shape and velocity.

Common Pitfalls and Considerations

Accurate calculations require attention to these details:

  1. Density variation with temperature — Fluid density changes with temperature. Water at 4°C is denser than at 20°C. If precision matters, use density at the actual temperature of your experiment, not reference values.
  2. Complete versus partial submersion — Archimedes' principle applies fully only when an object is completely submerged. Floating objects displace fluid equal to their weight, not their volume. Don't confuse apparent weight measurements between fully and partially submerged cases.
  3. Gravity variations matter at scale — Using 9.8 m/s² works for Earth's surface, but deep underwater or at high altitudes, gravitational acceleration shifts slightly. For engineering projects, use local gravity measurements rather than standard values.
  4. Compressible fluids and objects — Gases compress significantly under pressure, so buoyancy calculations become complex underwater at depth. For liquids and rigid objects in shallow water, incompressibility assumptions hold fine.

Determining Object Properties from Buoyancy

If you know an object's mass in air and its apparent mass when submerged, you can calculate its density and the displaced fluid volume. The difference between true and apparent mass equals the mass of displaced fluid.

Measure a rock at 540 g in air and 340 g in water. The 200 g difference is the water displaced. Since water has density 1000 kg/m³, the rock's volume is 200 g ÷ 1000 kg/m³ = 0.0002 m³. The rock's density is then 540 g ÷ 0.0002 m³ ≈ 2.7 g/cm³—indicating granite or similar silicate composition.

This method works for any fluid. Simply replace water's density with your fluid's density, and the calculation reveals the object's true volume and density, useful for identification and purity testing.

Frequently Asked Questions

Why do ships made of steel float when steel itself sinks?

Steel floats because a ship displaces far more water volume than a solid steel block of the same mass. The hull encloses air, dramatically increasing the average density of the whole vessel below that of water. A naval vessel's total weight divided by its submerged volume gives a density less than 1000 kg/m³, achieving buoyancy. The shape matters—spreading that steel across a large hull volume is the key.

How do submarines control their depth without active propulsion?

Submarines adjust buoyancy by flooding or purging ballast tanks with seawater. Adding water increases the vessel's average density, causing it to sink. Removing water decreases density, making it rise. At a specific ballast water volume, the submarine's weight exactly equals the buoyant force, achieving neutral buoyancy and hovering at constant depth. This equilibrium lets them cruise horizontally using only steering, not constant engine power.

Can Archimedes' principle predict whether an object will float before testing?

Yes. Compare the object's density to the fluid's density. If the object is less dense than the fluid, it floats. If denser, it sinks. If equal in density, it remains neutrally buoyant (neither rises nor sinks naturally). For example, ice at approximately 917 kg/m³ floats in water at 1000 kg/m³. Aluminum at 2700 kg/m³ sinks in water but floats in mercury at 13,600 kg/m³.

Why does a balloon feel lighter when held underwater despite being pushed down?

The buoyant force from displaced water reduces the balloon's apparent weight. A balloon displaces significant water volume relative to its mass, creating large upward buoyancy. Though you feel pressure pushing it downward (the water's weight above), the scale shows reduced force compared to air because buoyancy partially counteracts gravity.

How does salinity affect buoyancy in ocean water?

Saltwater is denser than fresh water—approximately 1025 kg/m³ versus 1000 kg/m³. This increased density generates stronger buoyant force on submerged objects. People float more easily in oceans and salt lakes (like the Dead Sea) because the buoyant force is larger. This is why swimmers experience noticeably different buoyancy between swimming pools and the sea.

What is apparent weight and why does it matter?

Apparent weight is what a scale reads when an object is submerged—always less than its true weight in air. A bathroom scale placed underwater measures the object's true weight minus buoyant force. This matters for calibration in laboratory precision balances submerged for measurements, medical imaging in water tanks, and understanding why astronauts feel weightless in water during training—buoyancy cancels gravity's pull.

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