Archimedes' Principle

Background Reading

Science historians generally agree that the idea for his principle occurred to Archimedes as he entered a bath pool. As he waded into deeper water, the force on his feet became less.

Archimedes' Principle is that an object totally or partially immersed in a fluid (liquid or gas) is buoyed (lifted) up by a force equal to the weight of the fluid that is displaced.

It has numerous applications, one of which is the determination of density and specific gravity. In the following discussion, the subscripts S, W, and A represent the Substance, Water, and Air, respectively.

Density is the mass per unit volume of a substance. rho = M / V

Specific Gravity is the ratio of the density of the substance to the density of water. SG = rhoS / rhoW.

The metric system unit GRAM is defined to be the mass of one cubic centimeter (one milliliter) of pure water at 3.98oC. Thus, for water rhoW = 1 gram/cm3 and, if errors due to impurities and/or temperatures are tolerable, this is a great convenience. The definitions given above suggest various methods for determining specific gravity.

SG = rhoS / rhoW = (MS / VS) / (MW / VW) = (MSg / VS) / (MWg / VW) = (WS / VS) / (WW / VW)

The definition of SG was applied for the first equality; the definition of density was applied for the second equality. For the third equality, the numerator and denominator were multiplied by "g". WS is the weight of the substance measured in air; WW is the weight of the water displaced by the substance when it is immersed.

In some circumstances, the volume of the substance is equal to the volume of the water. In particular, when a solid object is completely immersed in water, the volume of the water displaced must be equal to the volume of the object. Furthermore, by Archimedes' Principle, upon immersion the object would receive a buoyant force equal to the weight of the water displaced. Thus, an object weighed in air and then weighed while immersed in water would have an effective weight that was reduced by the weight of the water displaced, if the buoyant force of the air is negligible. When weighed in air, the object receives a buoyant force equal to the weight of the air displaced by the object. However, the density of air is small enough (compared to the density of most solids) to allow this buoyant force to be neglected when weighing most solids in air. (rhoair = 1.3 x 10-3 grams/cm3)

For an object more dense than water

SG = (WS / V) / (WW / V) = WS / WW = WS / (buoyant force) = WS / (loss of weight in water)
= WS / (WS - weight of substance in water)

This suggests a method for determining the SG of an object more dense than water; namely, weight an object in air and weigh it while completely immersed in water. The SG would then be the weight in air divided by the apparent loss of weight when weighed in water.

A device called a Jolly Balance is designed to measure the weight of objects in air and in water by reading the elongation of a spring. It utilizes a Vernier scale to read the elongation of the spring to the nearest 0.05 mm, and so is very precise. Hooke's law

F = - k x

should apply to the spring, where F is the force (weight) that stretches the spring by an amount x, and k is the spring constant. Substituting into the equation above, we find

SG = WS / (WS - weight of substance in water) = (k xA) / (k xA - k xW)

When the unknown spring constant is canceled, the SG may be found using only the two spring elongations

SG = xA / ( xA - xW)

For an object less dense than water

The equation above is only true if the object is more dense than water. If the object is less dense than water, a lead weight must be attached and three spring elongations must be measured to determine the SG. The three elongations are The latter two elongations are used to determine the loss of weight in water, that is, the denominator of the SG definition. You should derive this formula in terms of xA, xB, and xC.

SG = WS / (loss of weight in water) = ?

Having determined the SG for sinking and floating objects using Archimedes' Principle and the Jolly Balance, it is desirable to use an independent method for comparison. Since SG = rhoS / rhoW, and since rhoW = 1 gram/cm3), one can determine the density of the object by measuring its mass and volume directly.

For a liquid

The Volumetric Flask (or Pycnometer) has a hollow stem stopper that allows one to prepare equal volumes of fluids very reproducibly. If the mass of the flask is measured (1) when empty, (2) when filled with the fluid, and (3)when filled with water, the SG of the fluid can be determined. Since the volume is the same, it will cancel out of the SG fraction.


We will perform three separate experiments in which we will determine the density and specific gravity of
  1. a solid more dense than water
  2. a solid less dense than water
  3. a liquid

Set up the Jolly Balance

Part 1- an object more dense than water

Part 2 - an object less dense than water

Part 3 - a liquid

Error analysis

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