Understanding Archimedes’ Principle
Archimedes’ principle is a fundamental concept in fluid mechanics, describing how objects behave when immersed in a fluid. It states that a body submerged in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid. This principle explains why objects float, sink, or remain neutrally buoyant in liquids or gases.
Historically attributed to the ancient Greek mathematician and inventor Archimedes of Syracuse, this principle has wide-ranging applications in engineering, navigation, and scientific research. It helps determine the buoyant force acting on submerged objects and forms the basis for designing ships, submarines, hot-air balloons, and hydrometers.
Practical Applications of Archimedes’ Principle
- Submarines: Submarines utilize ballast tanks that can be filled with water or expelled with air to control buoyancy. By adjusting the mass within these tanks, submarines can submerge or surface, maintaining stability underwater based on Archimedes’ principle.
- Hydrometers: Instruments like hydrometers measure the specific gravity of liquids. They are calibrated to float at different levels depending on the liquid’s density, leveraging the principle that the buoyant force varies with fluid density.
- Hot-Air Balloons: Hot-air balloons float because heated air inside the balloon becomes less dense than the surrounding cooler air. By heating the air, the buoyant force exceeds the weight of the balloon, allowing it to ascend, illustrating buoyancy effects in gases.
The Mathematical Expression of Archimedes’ Principle
The core formula representing this principle is:
Fb = ρ × g × V
where:
- Fb = Buoyant force exerted on the object
- ρ = Density of the fluid
- g = Acceleration due to gravity
- V = Volume of fluid displaced by the object
Deriving Archimedes’ Principle
- The mass of the displaced fluid can be expressed as:
- Mass = Density × Volume = ρ × V
- Since density is defined as mass per unit volume,
- the weight of the displaced fluid becomes:
- Weight = Mass × g = ρ × V × g
- Therefore, the buoyant force, which is equal to this weight, can be written as:
- Fb = ρ × V × g
Sample Problems and Calculations
Example 1:
Calculate the buoyant force acting on a steel sphere with a radius of 6 cm when submerged in water. Assume:
- Radius (r) = 6 cm = 0.06 m
- Density of water (ρ) = 1000 kg/m3
- Gravity (g) = 9.8 m/s2
Solution:
- Volume of the sphere: V = (4/3)πr3
- V = (4/3)π(0.06)3 ≈ 9.05 × 10-4 m3
- Bouyant force: Fb = ρ × g × V
- Fb = 1000 × 9.8 × 9.05 × 10-4 ≈ 8.87 N
Example 2:
Determine the buoyant force on a floating object that is 95% submerged in water. The density of water is 1000 kg/m3.
Given:
- Density of water (ρ) = 1000 kg/m3
- Percentage of submersion = 95%
Approach:
- Since 95% of the object is submerged, the volume of displaced water is 0.95 times the total volume of the object.
- The buoyant force is then: Fb = ρ × g × V
- Rearranged, the volume of the displaced fluid is: V = (Fb) / (ρ × g)
Summary and Final Remarks
Archimedes’ principle provides a robust framework for understanding buoyancy phenomena. It explains how objects behave when immersed in fluids, whether they float, sink, or attain equilibrium. This principle not only underpins fundamental scientific concepts but also enables practical engineering solutions across diverse industries, from maritime navigation to aeronautics.
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