Archimedes' Principle states that any object wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. This fundamental principle explains why objects float or sink and is the basis for understanding buoyancy in liquids and gases. The principle applies to all fluids—liquids and gases—and is independent of the shape, size, or composition of the submerged object, depending only on the volume of fluid displaced.
Historical Context: The principle was discovered by Archimedes of Syracuse (287-212 BCE) while solving a problem for King Hiero II. According to legend, Archimedes realized the principle while taking a bath and observing the water displacement, famously shouting "Eureka!" ("I have found it!"). He used this discovery to determine if a royal crown was made of pure gold without damaging it, marking one of the first applications of non-destructive material testing.
Archimedes' Principle defines the properties of the buoyant force, a fundamental vector quantity in fluid mechanics that governs the behavior of objects in fluids.
| Property | Details |
|---|---|
| Nature | The buoyant force is a vector quantity. |
| SI Units | Newtons (N), as it is a type of force. |
| Magnitude | Equal to the weight of the fluid displaced by the object. It is calculated as the product of the fluid's density (ρ), the submerged volume of the object (V), and the acceleration due to gravity (g). |
| Direction | The buoyant force always acts vertically upward, opposing the force of gravity. |
| Point of Application | The force acts through the center of buoyancy, which is the geometric center (centroid) of the displaced volume of fluid. |
| Dimensional Formula | [M][L][T]⁻², the standard dimensions for force. |
The condition for an object's behavior in a fluid is determined by comparing its density to the fluid's density:
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| \( F_b \) | Buoyant Force | Newton (N) | The upward force exerted by the fluid on the submerged or floating object. |
| \( \rho_{fluid} \) | Fluid Density | kg/m³ | The mass per unit volume of the fluid in which the object is submerged. |
| \( V_{displaced} \) | Displaced Volume | m³ | The volume of the fluid that is moved out of the way by the object. For a fully submerged object, this equals the object's total volume. |
| \( g \) | Gravitational Acceleration | m/s² | The acceleration due to gravity, approximately 9.8 m/s² on Earth's surface. |
| \( \rho_{object} \) | Object Density | kg/m³ | The average mass per unit volume of the object. |
| \( V_{total} \) | Total Volume of Object | m³ | The entire volume occupied by the object. |
| \( V_{submerged} \) | Submerged Volume | m³ | The portion of the object's volume that is below the fluid's surface. |
| \( W \) | Weight | Newton (N) | The gravitational force on an object, calculated as \( W = mg \). |
Archimedes' Principle can be derived by considering the pressure difference a fluid exerts on a submerged object. Let's consider a rectangular object of height \( h \) and face area \( A \) fully submerged in a fluid of density \( \rho_{fluid} \).
1. The pressure on the top surface of the object, at a depth \( d_{top} \), is given by the hydrostatic pressure formula:
2. This pressure results in a downward force on the top surface:
3. Similarly, the pressure on the bottom surface, at depth \( d_{bottom} = d_{top} + h \), is greater:
4. This results in an upward force on the bottom surface:
5. The net force due to the fluid is the buoyant force \( F_b \), which is the difference between the upward and downward forces. The horizontal forces cancel out due to symmetry.
6. Simplifying the expression:
7. Since the volume of the rectangular object is \( V = h \times A \), which is also the volume of the displaced fluid \( V_{displaced} \), we arrive at Archimedes' Principle:
The principle's application results in three primary conditions for an object in a fluid, based on the relationship between the object's density and the fluid's density.
| Type / Case | Description | When to Use |
|---|---|---|
| Floating Object | An object floats when the buoyant force equals its weight, with only a part of its volume submerged. The net force on the object is zero. | Used when the object's average density is less than the fluid's density (ρ_obj < ρ_fluid). |
| Sinking Object | An object sinks when its weight is greater than the maximum possible buoyant force (when fully submerged). There is a net downward force. | Used when the object's average density is greater than the fluid's density (ρ_obj > ρ_fluid). |
| Neutrally Buoyant (Suspended) | An object remains suspended at a constant depth when its weight is exactly equal to the buoyant force when fully submerged. The net force is zero. | Used when the object's average density is equal to the fluid's density (ρ_obj = ρ_fluid). |
Naval Architecture: The principle is the foundation of ship and boat design. Engineers calculate the required hull volume and shape to displace enough water to support the vessel's total weight, including cargo and crew.
Submarine Technology: Submarines control their depth by altering their buoyancy. They take in water into ballast tanks to increase their overall density and sink (negative buoyancy), and expel water with compressed air to decrease density and rise (positive buoyancy).
Material Testing and Hydrometry: Archimedes' principle is used to determine the density of materials. By measuring an object's weight in air and its apparent weight when submerged in a fluid of known density, its volume and density can be calculated accurately. A hydrometer is an instrument that uses this principle to measure liquid density.
Aerospace Engineering: Lighter-than-air vehicles like hot air balloons and blimps operate on Archimedes' principle applied to air. A balloon rises because the hot, less-dense air inside it displaces a volume of colder, denser ambient air whose weight is greater than the total weight of the balloon.
Swimming in the Ocean vs. a Lake: People find it easier to float in the ocean than in a freshwater lake. This is because seawater is denser than freshwater due to its salt content. According to Archimedes' principle, the denser fluid provides a greater buoyant force for the same volume of displaced water, making it easier to stay afloat.
Life Vests: A life vest works by adding a large volume for very little mass. When worn, it significantly increases the total volume of the person, which increases the volume of water they displace. This creates a buoyant force large enough to overcome the person's weight, keeping their head above water even if they are unconscious.
Ice Cubes in a Drink: An ice cube floats in a glass of water because ice is about 9% less dense than liquid water. The ice cube sinks until it has displaced a volume of water that weighs exactly as much as the entire ice cube. This is why you see approximately 90% of the ice cube submerged below the water's surface.
| Quantity | Symbol | SI Unit | Dimensional Formula |
|---|---|---|---|
| Buoyant Force | \( F_b \) | Newton (N) | \( [M][L][T]^{-2} \) |
| Density | \( \rho \) | kilogram per cubic meter (kg/m³) | \( [M][L]^{-3} \) |
| Volume | \( V \) | cubic meter (m³) | \( [L]^3 \) |
| Gravitational Acceleration | \( g \) | meter per second squared (m/s²) | \( [L][T]^{-2} \) |
Dimensional Analysis of Archimedes' Principle: We can verify that the dimensions of the right side of the formula match the dimensions of force.
The primary formula is \( F_B = \rho_{fluid} V_{displaced} g \). It calculates the buoyant force (\( F_B \)), which is the upward force exerted by a fluid on any object wholly or partially submerged within it. This force is equal to the weight of the fluid displaced by the object.
In this equation, \( F_B \) is the buoyant force in Newtons (N), \( \rho_{fluid} \) is the density of the fluid in kg/m³, \( V_{displaced} \) is the volume of the fluid displaced by the object in m³, and \( g \) is the acceleration due to gravity, approximately 9.8 m/s².
You compare the object's density (\( \rho_{object} \)) to the fluid's density (\( \rho_{fluid} \)). If \( \rho_{object} < \rho_{fluid} \), the object will float because the buoyant force can exceed its weight. If \( \rho_{object} > \rho_{fluid} \), the object will sink because its weight is greater than the maximum possible buoyant force.
A common mistake is to think that the buoyant force depends on the object's weight or density. The buoyant force is determined solely by the weight of the displaced fluid, which depends on the fluid's density (\( \rho_{fluid} \)) and the submerged volume (\( V_{displaced} \)), not the object's characteristics.
Naval architecture is a direct application. A massive steel ship floats because its hull displaces a large volume of water, generating a buoyant force equal to the ship's total weight. Similarly, submarines use ballast tanks to take in or expel water, changing their average density to control their buoyancy and dive or surface.
Archimedes' Principle connects density to force. The buoyant force is a direct consequence of pressure increasing with depth in a fluid. By comparing the object's density to the fluid's density, we can determine the relationship between the object's weight (a downward force) and the buoyant force (an upward force), predicting its motion in the fluid.