Archimede's Principle
Archimede's principle states that an object immersed in a fluid is exerted upon by an upward bouyant force equal to the weight of the displaced fluid. The bouyant force is the force due to the difference between the upward pressure (on bottom surface) and the downward pressure (on top surface) acting on the object.
Let's consider a cylinderical object of density ρo height h and base area A immersed in a fluid of density ρf . The bouyant force (B) is equal to (P - Po )A where P is the pressure on the bottom surface and Po is pressure at the top surface. But P - Po = ρf |g|h. Therehore the bouyant force is given by B = Ahρf |g|h. The product Ah is equal to the volume V of the cylinder.
B = Vρf |g|
Since the product Vρf is equal to the mass of the fluid with the same volume as the object, it follows that bouyant force is equal to the weight of the displaced fluid.
An object immersed in a fluid weighs less than it does in air because of the upward bouyant force exerted by the fluid. The weight inside a fluid (Wf ) is equal to the difference between the weight in air (Wa ) and bouyant force.
Wf = Wa - B
Its weight in air i equal to the product of its mass ( mo ) and gravitational acceleration; and its mass is equal to the product of its density and its volume.
Wa = ρoV|g|
Now the weight in fluid can be expressed in terms of the densities by using expressions for the weight in air and bouyant force in terms of density.
Wf = ( ρo - ρf )V|g|
For a floating object, the weight of the object in air and the bouyant force exerted by the fluid must balance each other, because the object is in equilibrium. Suppose a floating object of density ρo and volume V floats in a fluid of density ρf with Vi part of its volume immersed. Its weight in air is equal to ρo V|g| and according to Archimede's principle ( bouyant force equals weight of displaced fluid ) the bouyant force is equal to ρf Vi |g|. Equating the weight in air and the bouyant force, the following equation for floating objects is obtained.
ρo ⁄ ρf = Vi ⁄ V
The following observations can be made from this equation: 1) an object whose density is smaller than the density of the fluid floats partially immersed 2) an object whose density is equal to the density of the fluid floats with all of its volume immersed. and 3) an object whose density is greater than that of the fluid sinks.
Fluid Dynamics
Fluid dynamics is the study of fluides in motion.
Continuity Equation
The Continuity equation is a mathematical statement of the fact that the amount of fluid that enters a tube is equal to the amount of fluid that leaves the tube in the same interval of time. Suppose fluid enters a tube of cross-sectional area A1 with speed v1 and leaves a tube of cross-sectional area A2 with speed v2. In a time interval Δt, fluid of length v1 Δt enters tube 1 and fluid of length v2 Δt leaves tube 2. In other words, in a time interval Δt, fluid of volume A1v1 Δt enters tube 1 and fluid of volume A2v2 Δt leaves tube 2. Equating these two volumes, the equation so called continuity equation is obtained.
A1v1 = A2v2
Bernoulli's Equation
Let's consider two tubes, tube 1 and 2, connected together with the elevation of the first being y1 and the elevation of the second being y2. There are two kinds of forces acting on a fluid flowing through these tubes. These are gravity and the force due to pressure difference in the tubes. Gravity is a conservative force but the force due to pressure difference is non-conservative. The work done by a non-conservative force is equal to change in mechanical energy. If a part of the fluid of mass Δm is taken, the change in mechanical energy is ( Δmv2 2 ⁄ 2 + Δm|g|y2 ) - ( Δmv1 2 ⁄ 2 + Δm|g|y1 ) and the work done by the force due to pressure difference is ( P1 - P2 )ΔV where ΔV is the volume of the fluid. Equating these two expressions, dividing the equation by ΔV and noting that Δm ⁄ ΔV is equal to the density of the fluid ρ, the following equation that is called Bernoulli's equation is obtained.
P1 + ρ|g|y1 + ρv1 2 ⁄ 2 = P2 + ρ|g|y2 + ρv2 2 ⁄ 2