Physics 111 Elementary Physics

Physics 111: Elementary Physics

Laboratory B

Archimedes Principle

1. Introduction

King Heiron of the Greek city-state of Syracuse, in Sicily, assigned to Archimedes (B.C. 287? 212) the problem of determining whether a goldsmith may have substituted some silver for some of the gold which the goldsmith had been given to make a crown for the king. There seemed to be no way to determine this short of melting down the crown. One day, while bathing, Archimedes realized that any object, however irregularly shaped, will displace a volume of fluid equal in volume to that of the object which is immersed in the fluid. It was then an easy matter to compare the volume of the crown with the volume of equal masses of pure gold and pure silver, and to find whether the crown was short in mass. King Heron was pleased with Archimedes, but not with the goldsmith.

Archimedes also found that the buoyant force exerted by a fluid on an object floating or immersed in the fluid is equal to the weight of the fluid displaced. This latter fact, now known as Archimedes Principle, is of central importance when designing ships and boats. One can make vessels such as rafts or hollowed-out logs of materials themselves light enough to float. One can make vessels capable of carrying much larger loads of much stronger materials, such as metals which are too dense to float, by using the materials to make a framework over which is placed a watertight covering. The volume thus defined is great enough to cause displacement of large enough a volume of water to bear up the load to be carried in the vessel.

This experiment will apply Archimedes Principle to determine the relative density of three pieces of metal. The weight of a piece will be determined in air, Wa, and when it is immersed in water, Wi (see figure at right). The buoyant force, B, which the water exerts on the piece of metal then is given by

B = Wa Wi

(1)

which is also equal to the weight of the water displaced by the metal. Note that Wa = mg.

The volume of the water displaced is equal to the volume, V, of the piece of metal. If the density of the metal is , and that of the water is w, then

= Wa/(Vg), (2)

and

w = B/(Vg).

(3)

In Eqs. (1) and (2), the symbol g represents the acceleration due to gravity ( g = 9.80 m/s2). The mass m, of an object is given by m = Wa/g. The relative density (or specific gravity) of the metal is the ratio of its density to that of water,

Relative density = /w = Wa/B = Wa/(Wa Wi) = ma/(ma mi).

(4)

2. Procedure

A. For each piece of metal it will be necessary to use the balance to determine the mass of the piece first in air, and then when the piece is immersed in water. In preparation for the latter, a piece of thread should be used to suspend the piece of metal from the place on the balance where the balance pan is usually suspended. Enough room should be left that a beaker of water can be moved under the piece of metal. After the mass of the piece of metal in air has been found, move the beaker of water under the metal and raise the beaker until the metal is immersed and record the new reading, mi. Enter the data in the table and use Eq. (4) to calculate the relative density. Use a table of relative densities of common materials to identify the type of metal in each piece.

B. One of the pieces of metal will be a regular cylinder. The volume of the cylinder is given by the product r2h, where r is the radius of the cylinder and h is the height (or length) of the cylinder. Use the caliper to measure the radius of the cylinder and its height. Use these data to calculate a theoretical value, Vtheo, of the volume of the cylinder. The values of the masses of the cylinder which are recorded above can be used to determine the buoyant force, B. The fact that the density of water is taken to be 103 kg/m3 (1 g/cm3) can then be combined with Eq.(3) to determine a value of the volume Vexp. Vexp is the V which appears in equations (2) and (3). Combining (3) with (1) we find that

Vexp = (Wa Wi)/(wg) = (ma mi)/w

(5)

Find both Vtheo and Vexp, and determine the percentage difference of Vexp from Vtheo,

V/Vtheo = (| Vexp Vtheo |)*100 /Vtheo.Physics 111: Elementary Physics

Pre-Lab Exercise

Archimedes Principle

Name: _____________________________ Section: ____

1. With reference to the equilibrium of the piece of metal justify the form of equation (1).

2. Verify the last equality in Eq. (4).

3. Some students performed the experiment with a certain kind of metal. They found for the masses the following values. Determine the relative density of the metal. Use the table of densities in your text book to identify the metal.

ma

mi

Relative Density.

Type of metal

79.2 g

70.1 g

___________

__________

4. The same students also have obtained values for r, h and Vexp. Complete the data table. Give careful attention to units and significant figures.

r h Vtheo

Vexp

V/Vtheo(%)

0.695 cm 5.95 cm _______ ________ __________

5. State briefly (100 words or less) the objectives of this experiment and the principle which is to be tested.Physics 111: Elementary Physics

Lab Report

Archimedes PrincipleInvestigators:________________________ , _______________________

________________________ , _______________________

________________________

Date: _____________

Procedure: Describe briefly (200 words or less) the procedures used in this experiment.

Data:Procedure A

ma

mi Relative DensityType of metal

1.

________ ________ _________ __________

2.

________ ________ _________ __________

3.

________ ________ _________ __________

Procedure B

r h Vtheo

Vexp

V/Vtheo(%)

______ ______ _______ _______ _________

Discussion: Discuss the outcome of the experiment, including estimates of the accuracy of relative densities found with those in standard tables, and the comparison of Vtheo with Vexp.