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, W a, 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 = W a – 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 ρ, and
that of the water is ρw, then
ρ = Wa/(Vg),
(2)
and
ρw = B/(Vg).
(3)
In Eqs. (1) and (2), the symbol g represents the acceleration due to gravity ( g = 9.80 m/s 2). 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 = ρ/ρw = 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 лr2h, 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)/(ρwg) = (ma – mi)/ρw
Find both Vtheo and Vexp, and determine the percentage difference of Vexp from Vtheo,
ΔV/Vtheo = (| Vexp – Vtheo |)*100 /Vtheo.
(5)
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.
4.
ma
mi
79.2 g
70.1 g
Relative Density.
Type of metal
___________
__________
The same students also have obtained values for r, h and V exp. Complete the data table. Give
careful attention to units and significant figures.
r
0.695 cm
h
5.95 cm
Vtheo
_______
Vexp
________
ΔV/Vtheo(%)
__________
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’ Principle
Investigators:
________________________ ,
_______________________
________________________ ,
_______________________
________________________
Date: _____________
Procedure: Describe briefly (200 words or less) the procedures used in this experiment .
Data:
Procedure A
ma
mi
Relative Density
Type of metal
1.
________
________
_________
__________
2.
________
________
_________
__________
3.
________
________
_________
__________
Procedure B
r
______
h
______
Vtheo
Vexp
ΔV/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 V theo with Vexp.