The buoyancy force on a cylinder is equal to the weight of the fluid displaced by the cylinder, and can be expressed as FB = Vcylinder ρ g, where Vcylinder is the volume of the submerged part of the cylinder, ρ is the density of the fluid, and g is the acceleration due to gravity.
Understanding Buoyancy
Buoyancy is an upward force exerted by a fluid that opposes the weight of an immersed object. This force arises because the pressure in a fluid increases with depth. Therefore, the pressure at the bottom of the submerged object is greater than the pressure at the top, resulting in a net upward force.
Calculating Buoyancy Force for a Cylinder
To calculate the buoyancy force on a cylinder, we need to know:
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The volume of the submerged part of the cylinder (Vcylinder): This is the portion of the cylinder that is below the surface of the fluid. If the cylinder is fully submerged, this is simply the total volume of the cylinder. If the cylinder is only partially submerged, you'll need to calculate the volume of the submerged portion. The volume of a cylinder is given by V = A * h, where A is the cross-sectional area of the base and h is the height (or depth) of the submerged portion.
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The density of the fluid (ρ): This is the mass per unit volume of the fluid. Common units are kg/m³ or g/cm³.
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The acceleration due to gravity (g): This is approximately 9.81 m/s² on Earth.
The buoyancy force is then calculated using the formula:
FB = Vcylinder ρ g
Example
Let's say we have a cylinder with a radius of 0.1 meters and a height of 0.5 meters. The cylinder is made of wood and is floating in water. Only 0.2 meters of the cylinder's height is submerged.
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Calculate the submerged volume (Vcylinder):
- Area of the base (A) = π r² = π (0.1 m)² ≈ 0.0314 m²
- Submerged height (h) = 0.2 m
- Vcylinder = A h = 0.0314 m² 0.2 m ≈ 0.00628 m³
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Determine the density of water (ρ):
- The density of water is approximately 1000 kg/m³.
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Use the acceleration due to gravity (g):
- g = 9.81 m/s²
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Calculate the buoyancy force (FB):
- FB = Vcylinder ρ g = 0.00628 m³ 1000 kg/m³ 9.81 m/s² ≈ 61.6 N
Therefore, the buoyancy force acting on the cylinder is approximately 61.6 Newtons.
General Formula & Derivation
The more general derivation, as indicated by the provided reference, assumes that the cylinder is oriented vertically and partially submerged. If h1 is the depth of the bottom of the cylinder, and h2 is the depth of the top of the cylinder (both measured from the surface of the fluid), then the height of the submerged portion is (h1 - h2). If A is the cross-sectional area of the cylinder, then the buoyant force can be calculated as:
*FB = (h1 - h2) ρ g A**
This is mathematically equivalent to the earlier equation, as (h1 - h2) * A represents the volume of the submerged portion.
Conclusion
The buoyancy force acting on a cylinder is determined by the volume of the submerged portion, the density of the fluid, and the acceleration due to gravity. Understanding this relationship is crucial in various fields, including naval architecture, fluid mechanics, and even everyday observations of floating objects.
