Fluid Mechanics Applications/B10: Archimedes Principle

Archimedes Principle
When a body is immersed fully or partially in a fluid, it experiences a buoyant force in upward direction which is equal to the weight of the fluid displaced by the body. Archimedes' principle is a law of physics fundamental to fluid mechanics.It was given by archimedes of syracuse.The principle applies to both floating and submerged bodies and to all fluids, i.e., liquids and gases. It explains not only the buoyancy of ships and other vessels in water but also the rise of a balloon in the air and the apparent loss of weight of objects underwater. It will help to determine whether a body placed in a liquid will float or it will sink.

Description
Experimentally seen, the Archimedes principle permits us to measure the volume of an object by measuring the volume of the fluid it displaces after submerging in liquids, and hence the buoyancy of an object immersed in a fluid can be calculated.

for an immersed body, the volume of the fluid displaced is equal to volume of submerged portion.E.g., by submerging a sealed 1-liter container of pepsi and when we displace 1-liter volume of the fluid, regardless of the containers content will be same.

An empty bottle when released in air will fall down due to force of gravitation. But if the same bottle when released under a fluid of density greater than air say water will be pushed upwards, towards the surface of water under the action of same gravitatonalforce.The extra force that comes in picture is the upthrust or called archimedes force.

Formula
For a fully submerged object, Archimedes' principle can be reformulated as follows:
 * $$\text{apparent immersed weight} = \text{weight of object} - \text{weight of displaced fluid}\,$$

then inserted into the quotient of weights, which has been expanded by the mutual volume


 * $$ \frac { \text{density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight of displaced fluid} }$$

yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes:


 * $$ \frac { \text {density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight} - \text{apparent immersed weight}}.\,$$

(This formula is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing.)

Archimedes Law of Floatation
Symbols used:-

M= mass of immersed object

a=acceleration of body in medium

g= acceleration due to gravity

V=volume of object when totally submerged

v=volume of object when partially immersed

$$\sigma$$=density of object

$$\rho$$=density of fluid

U= upthrust force When a body is immersed in a fluid then if :
 * The weight W of body is more than the up – thrust U ( W> U ), the body will sink, with acceleration as given below:-

$$Ma=Mg-u

\Rightarrow Ma=V \sigma g -V \rho g

\Rightarrow a=g(1-\rho/\sigma)$$ $$Mg=U \Rightarrow V \rho g=V \sigma g \Rightarrow \rho=\sigma$$ $$Mg=U \Rightarrow V \sigma g=v \rho g \Rightarrow V \sigma=v \sigma $$
 * The weight W of the body is equal to up – thrust U (U = W), the body floats with complete submergence.
 * The weight ( W< U$$$$), the body will float with some of its part outside the liquid.

Apparent loss in weight
According to the principle when a body is immersed in a fluid partially or wholly,  a part of its weight appears to lose which is equal to the displaced weight of the fluid..

Apparent weight of body = Actual Weight of Body – Upthrust $$= [Mg - M / \sigma \rho g] = Mg [1 - \rho / \sigma]$$

Applications Regarding Submarines
How They Work – Whether, the floating or submerging of a submarine is controlled by ship’s buoyancy which is controlled by the ballast tanks, which are found between the inner and outer hull of the submarine. When the ballast tanks are full of air, it is less dense than water around it andthus it has a positive buoyancy. Hence submarine will float.To submerge the submarine, sea water coming in through flood ports forces air out of the vents that are located on the top of the ballast tanks and the submarine began to sink, as now the submarine’s buoyancy is negative. By controlling the ratio of air to water, required depth can be achieved. When the weight of the submarine is equal to the amount of fluid displaced it achieve neutral buoyancy andhence will never rise or sink.To rise again the submarine, the compressed air is blown in the ballast tank which blow out the water and ship gains a positive buoyancy and become less dense than surrounding air hence rises.

Applications Regarding Hot Air Balloon
In the hot air balloon, the basic principle involved is the creation of buoyant force by the use of hot air. A hot air balloon consist of a basket which is suspended from a large bag called envelope. The air inside the bag is continously heated by a burner that sits in a basket through an opening. Due to this, the air inside the envelope becomes less dense than the ambient (cool)air. As a result of which, the balloon is lifted off the ground due to the buoyant force created by the surrounding air. If the balloon is to be lowered, the firing of the burner is stopped causing the hot air in the envelope to cool (hence decreasing the buoyant force), or a vent at the top of the bag is opened through which the hot gases escape(decreasing the buoyant force) causing the hot air balloon to descend. To maintain a constant height ,the burner is operated in a pulsating manner, causing the balloon to rise and fall. That’s how an approximate altitude can be maintained, as generating a zero buoyant force is practically impossible. If the balloon is to be moved In horizontal direction, then the wind direction must be known as it varies accordingly to the altitudes. So it is simply ascended or descended corresponding to the wind direction that the balloon is to be moved.