NMU-Physics
Archimedes Principle
[1]
Archimedes Principle
Materials: Mass balance in lab stand configuration, two metal masses, one wooden object, a lead or
other metal sinker, one bag of pellets, large metal cans and bench stands, string, and micrometer
1
Purpose
The goal of this laboratory is to investigate the density of different materials using Archimedes’ buoyancy
principle.
2
Introduction
The Greek mathematician, physicist, and inventor Archimedes (287 BC - 212 BC) developed his buoyancy
principle while supposedly taking a bath. The true root of this story is irrelevant. What matters here is
that an object submersed in water experiences a buoyant force, B, equal (in magnitude) to the weight of
the fluid which is displaced, Wdisplaced .
B = Wdisplaced
(1)
This buoyant force can then be used to determine the density or volume of the object which is immersed
in water (depending on the final goal). This allows the investigator to determine the volume and density
of irregularly shaped objects without damaging the object. Here is how it works.
Mass Balance
Water line
Object
Figure 1: Schematic of the Archimedes setup for an object more dense than water.
If an object is suspended from a mass balance so that the balance is at zero, then the tension in the
string is equal to the weight of the object. Now let the object be submerged in a tub of water (Fig. 1).
The free body diagrams for this scenario are shown in Fig. 2. In both cases the system is in equilibrium.
Therefore, summing the forces for Fig. 2A yields
ΣF = T 0 + B − Wdry = 0
(2)
where B is the buoyant force and Wdry is the weight of the object when measured in dry air. Looking at
Fig. 2B the sum of all forces yields
ΣF = T 0 − Wwet = 0
(3)
where Wwet is the weight of the object when suspended in water. Solving Eq. 3 for T 0 , substituting into
Eq. 2, and solving for B yields
B = (mdry − mwet )g.
(4)
NMU-Physics
Archimedes Principle
1
0
B
0
1
0
1
0
1
T’
0
1
0
1
0
1
11
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0W Dry
1
0110
1010T’
1010
011010
1010
1010
1010
10 W Wet
(A)
(B)
[2]
Figure 2: (A) Free Body Diagram for an object suspend in water indicating the buoyant force. (B) Free
Body Diagram for an object suspend in water indicating the apparent weight (wet weight).
Archimedes principle states that the buoyant force is equal to the weight of the fluid displaced by the
object (Eq. 1). Setting Eq. 4 equal to Eq. 1 yields
mf luid = (mdry − mwet ).
(5)
By substituting for the mass of the fluid the fluid’s density times volume and solving for the volume we
get
(mdry − mwet )
V =
(6)
ρf luid
where the fluid in this case is water. The volume of the fluid displaced is the same as the volume of the
object as long as the object is completely submerged.
This works well for the case in which the object is more dense than the fluid and will sink on its own.
For objects which are less dense than the fluid and float in the fluid, a sinker is necessary to completely
immerse the object in the fluid. Figure 3 shows the object in the “dry mass” setup and Fig. 4 shows the
object in the “wet mass” setup.
Mass Balance
Object
Water line
Sinker
Figure 3: Schematic with the floating object suspended in dry air with a sinker attached.
Keeping the sinker submerged at all times is important to this experiment. Looking at the FBD’s for
the sinker/object system as a whole (see Figure 5, it can easily be seen that the weight of the sinker is in
both diagrams and is the same in both cases. By keeping the sinker submerged for both measurements,
the weight of the sinker cancels in the force analysis. This can easily be shown by first summing the
forces on the two diagrams to get the following two equations.
ΣF = T 0 + B − Wdry − Wsinker = 0
(7)
NMU-Physics
Archimedes Principle
[3]
ΣF = T 0 + B − Wwet − Wsinker = 0
(8)
Eliminating T 0 from the two equations yields the same result as was found previously in Eq. 6.
Mass Balance
Water line
Object
Sinker
Figure 4: Schematic with the floating object submerged.
1
0
B
0
1
0
1
0
1
T’
0
1
0
1
0
1
11
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0W Dry
1
W Sinker
(A)
1
0
0
1
T’
0
1
0
1
0
1
0
1
0
11
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0 W Wet
1
W Sinker
(B)
Figure 5: Free Body Diagrams for the sinker/object system. Note that the FBD’s treat the sinker and
object as one system. (A) The FBD when the object is suspended in the air. (B) The FBD when the
object is submerged.
3
Procedure
1. Collect all necessary information about the object. Diameter and length if the object is regularly
shaped.
2. Attach the object to the string suspended under your balance.
3. Measure the mass of the object while suspended in air. This is your “dry mass”.
4. Submerse the object in water and measure the mass again.
5. Repeat the measurement for the second metal cylinder and the bag of pellets.
6. For the wooden object, measure the mass, diameter and length as necessary.
7. Attach the sinker to the wooden object and suspend the system below the balance as in Fig. 3.
Remember that in order for this to work the sinker must be below the water line and the wood
object above the water line.
NMU-Physics
Archimedes Principle
[4]
8. Measure the mass of the system. This is the “dry mass” of the wooden object.
9. Submerge the wooden object and then mass the system again (Fig. 4). This is the “wet mass” of
the wooden object.
10. Clean up the area and pour all the water down the drain.
4
Analysis
IMPORTANT NOTE: For all four objects, show ALL calculations on a separate piece of paper. Report
the results in a clear fashion. Gibberish and scratch work will receive little or no credit.
1. For each object, calculate the volume of the object via the appropriate geometrical formula and
by Archimedes method. For the pellets you will only have Archimedes method since they are
irregularly shaped.
2. Calculate the density in units of g/cm3 for each object. Do this for each volume you calculated
in (1) above. Which mass measurement should you use for the density calculation? Answer this
question to yourselves before making each density calculation.
5
Questions
1. Using your textbook (or appropriate Internet resources, if needed), determine the primary type of
material for each object. You may need to consider “context” clues as well. (color, location, etc.)
2. Record the accepted value of the density you have identified for each object and your reputable
source of information. If you do not cite a reputable source you will not receive credit. Note that
‘wikipedia’ is not considered a reputable source.
3. Now compare the accepted value for the density to your measured values of the density. Do one
comparison for EACH value of the density. Use the following formula.
% Difference =
ρmeasured − ρaccepted
× 100 %
ρaccepted
References
• OpenStax College, College Physics, 21 June 2012 <http://cnx.org/content/col11406/latest>.
• Physics, 5th Edition, Cutnell and Johnson, Chapter 11.
• Previous NMU laboratory handout
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