Archimedes' law in pressure and buoyancy

Archimedes' Principle

Definition: Archimedes' principle states that a body submerged in a fluid (liquid or gas) experiences a buoyant force equal to the weight of the fluid displaced by the body.

Buoyant Force

Formula: ${𝐹}_{𝑏}=𝜌\cdot 𝑉\cdot 𝑔$ where:

• ${𝐹}_{𝑏}$ is the buoyant force.
• $𝜌$ is the density of the fluid.
• $𝑉$ is the volume of the fluid displaced by the body.
• $𝑔$ is the acceleration due to gravity.

Key Concepts

1. Buoyant Force:

   • An upward force exerted by a fluid that opposes the weight of an object immersed in it.
   • It acts through the center of buoyancy, which is the centroid of the displaced fluid volume.

2. Displacement:

   • When an object is placed in a fluid, it pushes the fluid out of the way and takes its place, displacing a volume of fluid equal to the volume of the object submerged.

3. Floating and Sinking:

   • An object will float if its density is less than the density of the fluid.
   • An object will sink if its density is greater than the density of the fluid.
   • An object will be neutrally buoyant if its density is equal to the density of the fluid.

Applications

1. Ships and Submarines:

   • Ships float because their overall density, including the air inside them, is less than the density of water.
   • Submarines adjust their buoyancy by changing the volume of water in their ballast tanks, thus changing their overall density.

2. Hydrometers:

   • Devices used to measure the density of a liquid by observing the level at which the device floats in the liquid.

3. Hot Air Balloons:

   • Balloons float in the air because the hot air inside the balloon is less dense than the cooler air outside.

Archimedes' Law in Pressure and Buoyancy

Hydrostatic Pressure: Hydrostatic pressure in a fluid increases with depth due to the weight of the fluid above.

Formula: $𝑃={𝑃}_0+𝜌\cdot 𝑔\cdot ℎ$ where:

• $𝑃$ is the pressure at depth $ℎ$.
• ${𝑃}_0$ is the atmospheric pressure on the surface.
• $𝜌$ is the density of the fluid.
• $𝑔$ is the acceleration due to gravity.
• $ℎ$ is the depth of the fluid.

Archimedes' Principle and Hydrostatics

• The buoyant force can also be derived from the pressure difference exerted by the fluid on the top and bottom surfaces of the submerged object.
• The pressure at a deeper point in the fluid is higher, creating a net upward force on the object.

Example Problems

1. Floating Object:

   • Consider a wooden block floating in water. To determine how much of the block is submerged, you need to compare the weight of the block with the buoyant force.

2. Sinking Object:

   • Consider a metal object sinking in water. The weight of the object is greater than the buoyant force, causing it to sink. The apparent weight of the object is the actual weight minus the buoyant force.

Summary

Archimedes' principle is fundamental in understanding buoyancy and hydrostatics. It explains why objects float or sink in fluids based on their density relative to the fluid and the buoyant force acting on them. This principle has widespread applications in designing ships, submarines, hydrometers, and hot air balloons, and in various scientific and engineering fields involving fluid mechanics.