IEMDaily - Video Lecture Notes

Law

Proof of Archimedes Principle

Consider a cylindrical body of cross section area

A

immersed in a fluid of density

ρ

as shown in the attached figure.
Force on the top surface is given by:

F_{1} = (P + h_{1} ρ g) A

downwards
Force on the bottom surface is given by:

F_{2} = (P + h_{2} ρ g) A

upwards
Net upward force due to buoyancy on the fluid column is given by:

F_{b} = F_{2} - F_{1} = (h_{2} - h_{1}) ρ g A

Mass of the displaced fluid is equal to the volume of displaced fluid times density of the fluid. Hence, weight of the displaced fluid is given by:

W = ρ V g

W = ρ A (h_{2} - h_{1}) g

From above equations,

W = F_{b}

Hence, upward buoyancy force on a body immersed in a fluid equals the weight of the fluid displaced by it. This is the statement of Archimedes Principle.

Example

Hydrometer- Principle of working

Hydrometer is an instrument used to measure the relative density of a liquid (w.r.t. water). Its operation is based on Archimedes' Principle and Principle of flotation. It has several uses like:

Lactometer (relative density of milk).
Saccharometer (measuring density of sugar in a liquid).

Example

Hydrometer- Construction

A hydrometer is usually made of glass, and consists of a cylindrical stem and a bulb weighted with mercury or lead shot to make it float upright. The liquid to test is poured into a tall container, and the hydrometer is gently lowered into the liquid until it floats freely. The length of hydrometer submerged is a function of specific gravity. Hydrometers usually contain a scale inside the stem, so that the person using it can read specific gravity. A variety of scales exist for various applications.

Example

Battery hydrometer

A battery hydrometer is an application of hydrometer and is based on Archimedes' Principle. The state of charge of a lead-acid battery can be estimated from the density of the sulfuric acid solution used as an electrolyte in the battery.

Example

Lactometer

A lactometer is an application of hydrometer and is used to check purity of cow's milk. It works on the principle of Archimedes' principle. The instrument is graduated into a hundred parts. Milk is poured in and allowed to stand until the cream has formed, then the depth of the cream deposit in degrees determines the quality of the milk. If the milk sample is pure, then the lactometer floats on it and if it is adulterated or impure, then the lactometer sinks.Note: The relative density of milk does not give a conclusive indication of its composition since milk contains a variety of substances that are either heavier or lighter than water.

Example

Archimedes principle in accelerated frame

Example:
A body having volume

V

and density

ρ

is attached to the bottom of a container as shown. Density of the liquid is

d

(> ρ)

. Container has a constant upward acceleration

a

. Find the tension in the string.
Solution:
Note that the acceleration has an effect on both upthrust and weight
Weight experienced by the mass =

ρ V (g + a)

Upthrust experienced by the mass =

d V (g + a)

Tension experienced by the string =

U p t h r u s t - W e i g h t

V (g + a) (d - ρ)

Definition

Explanation of Physical situations using Archimedes Principle

When any boat displaces a weight of water equal to its own weight, it floats. This is often called the "principle of flotation": A floating object displaces a weight of fluid equal to its own weight. Every ship, submarine, and dirigible must be designed to displace a weight of fluid at least equal to its own weight. A 10,000-ton ship must be built wide enough to displace 10,000 tons of water before it sinks too deep in the water.

Definition

Principle of Floatation

According to the Principle of Floatation, a body floats in a liquid if the weight of the body is equal to the weight of the liquid displaced by it.
Note: The floating of a body occurs if and only if the density of the body is less than or equal to the density of fluid.