-1 -
Physics 104---Lab 2:
Archimedes’ Principle
© 1994-2009, James J. DeHaven, Ph.D.
The legend goes that Archimedes was asked by the King of Syracuse (in Sicily, not New
York) to figure out whether the crown he wore was made of pure gold or not. Since he was unable
to find the volume, he did not know how to ascertain the density. The answer, to weigh the crown in
water, is said to have occurred to him in the public baths, from which, in his joy, he exited shouting,
“Eureka!”. (I’ve got it!)
In the process of articulating the problem, and thinking through its solution, Archimedes
stumbled upon the principle which bears his name today:
If an object is submerged in a fluid, the fluid supports the object with a force
equal to the weight of fluid displaced by the object.
We call the force that supports an object the buoyant force on the object.
• If the weight of a submerged object is greater than the buoyant force, the object sinks.
• If the weight of a submerged object is less than the buoyant force, the object will rise to the
surface of the fluid.
• If the weight of a submerged object is exactly equal to the buoyant force, the object will
hang motionless in the fluid.
• A floating object submerges a fraction of its volume so that the buoyant force exactly
equals the weight of the object.
In today’s lab you will explore the physics of fluids in the following four areas:
1) A direct confirmation of Archimedes’ Principle.
2) Determination of the density of a metal.
3) Determination of the density of a substance less dense than water.
4) Determination of the density of a liquid
Method
In order to complete today’s lab, you will need two capabilities:
1) You will need to be able to weigh objects under water;
2) You will need to be able to accurately determine volumes.
Volume will be obtained by measuring the water displaced by the test object in a graduated
cylinder.
Weighing will take place on a conventional triple-beam or electronic balance which has been
outfitted to weigh hanging masses. If you are using a mechanical balance, the fluid will be placed in
a beaker on the swing-out support (see figure 1). The mass to be weighed will be suspended from
the lower hook. The balance pan will hang freely from the upper hook as shown in figure 1.
-2 -
Balance Beam
Metal Sample
Support Arm
Balance Pan Swings Freely
Fig. 1 Mechanical balance setup for weighing an object submerged in a fluid.
If you are using an electronic balance, the fluid will be placed in a beaker on the plastic
beaker support, with the balance pan removed (see figure 2). The mass to be weighed will be
suspended from a wire hook.
-3 Balance
Hook
String
metal sample
Support Stand
Electronic
Balance
Fig. 2 Electronic balance setup for weighing an object submerged in a fluid. Note that
the balance pan is removed and replaced with the beaker support stand
Experimental
A. Experimental Verification of Archimedes’ Principle
1. Measure the mass of the cylindrical brass sample
2. Measure the volume of the cylindrical brass sample. You will need to use an electronic
balance to accomplish this. Use a graduated cylinder to measure the volume. Carefully place the
sample in a cylinder and add water until it is completely submerged. With an eyedropper, add water
until the level exactly reaches some easy-to-read reference value. Remove the weight and shake any
water drops back into the cylinder. Weigh the cylinder, fill back up to the reference mark, and weigh
again. The difference between the two masses can be used to deduce the mass of the water which
has been displaced by the hanging weight. Since the density of water is 1 g/mL, the number of mL
of water displaced equals the difference between the two weights.
-4 3. Now weigh the cylindrical brass sample while it is suspended in water using the setup
shown in Fig. 1 or Fig. 2 (depending on your balance).
4. According to Archimedes’ Principle, the buoyant force should equal the weight of the
water displaced by the object. To understand the idea of the buoyant force, consider the following
free body diagram for an object submerged in some fluid:
FB
(Bouyant Force)
Figure 3: Free Body Diagram
for a Submerged Object
mog
(Weight)
The buoyant force, FB, is the force that the fluid exerts on the object in order and that pushes the
object up. Whether the object will sink or rise to the surface will depend on the direction of the
NET FORCE. We write an expression for the net force, by examining figure 2 above:
Fnet = mog – FB
[1]
The behavior of the object will depend on whether the net force is greater than, less than, or equal to
zero:
[2]
If Fnet = 0
Object neither rises nor sinks
If Fnet > 0
Object sinks
If Fnet < 0
Object rises to surface
-5 In this experiment, you weigh an object that is suspended under water. What you are really
measuring when you weight this object is the net force in equation 1. We call this weight
THE APPARENT WEIGHT, w’ :
[3]
w' = m'o g
Actually, when you use a balance you get your result in terms of mo’, the APPARENT MASS of
the object. This doesn’t mean that the object has a new, different mass when it is under water; mass
is mass. It means that it feels or appears to have a smaller mass, and we call this smaller mass the
apparent mass.
Since the net force is equal to the apparent weight, we can rewrite equation [1] as follows:
[4]
w' = mog – F
B
But the buoyant force is just equal to the weight of the
fluid displaced (wf ) as shown in equation [5].
[5]
FB
Substituting [5] into [4] gives [6]
[6]
w' = mog – w f
[7]
m'o g = mog – mf g
= wf
Then use weight = mg to get [7].
[8] m'o = m o – m f
Cancel out ‘g’ to get [8] and now substitute the
definition of density to get [9].
[9] m'o = ρoV o – ρ V
f f
Because m= ρV
[10] V o = V
f
But we know that the volume of the object is the same
as the volume of the fluid that the object displaces.
Equation 10 simply states this.
Substituting [10] into [9] gives an expression which predicts the reading we should get on the
balance according to Archimedes Principle:
[11] m'o = V o ( ρ – ρ )
o
f
Compare your measured value for the apparent mass to that predicted by Archimedes’ principle.
-6 B. Measurement of Densities Using Archimedes’ Principle
1. You will use the density of the brass cylinder which you computed from the data you
took in part A.
2. Using Archimedes’ Principle determine the density of
a) the aluminum cylinder,
b) the plastic cylinder (see special directions below)
c)the irregular metal object using Archimedes’ principle.
Do not directly measure their volumes. Instead submerge the objects under water (note: use
the special directions below for the plastic object) and use the weight thereby obtained to determine
the densities. Recall that the actual mass of the object, mo, is given by:
m o = ρoV o
[12]
while the mass that we measure while the object is submerged, mo’ , the apparent mass, is given by
equation [9]:
m'o = ρoV o – ρf Vf
[9]
The difference between these two masses can be obtained simply by subtracting equation
[9] from equation [12]:
[13]
m o – m'o
= ρf Vf
Use [13] and [9] to relate the ratio of the masses to the ratio of the densities as follows:
mo
[14]
m o – m'o
=
ρoV o
ρf Vf
We then solve [14] for ρo , the density of the submerged object:
[15]
ρo = ρf
(
mo
m o – m'o
)
-7 Thus, since we know the density of the fluid (water) the densities of each of the three
materials can be determined. In each case, you need mo (the mass of the object), and mo’ (the
apparent mass of the object when it is weighed under water) for the sample.
Special Case of Plastic
In the case of the plastic object you will have to weigh it down to keep it under water. This means
that you can’t directly measure its apparent mass--you have to deduce it from things that you can
measure.
Use the brass weight (a rectangular solid mad of brass is provided for this purpose) to
weigh down the plastic cylinder:
Together, the brass weight and
plastic cylinder will have a
combined apparent mass, m', when
weighed under water.
Tie the brass and plastic weights
together and then suspend the
assembly and weigh under water.
Brass Weight
Plastic Cylinder
Figure 4: Brass weight and plastic
cylinder tight together with a string
The measured apparent mass, m’, is just the sum of the contributions from the brass weight
and from the plastic cylinder:
[16]
m'
= m'BW + m'P
-8 The problem here is that we don’t know mp’ , the apparent mass of the plastic. However, we
can easily find mBW’ , simply by weighing the brass weight under water separately. We know m’
from measuring the apparent mass of the brass weight and plastic together, so it is easy to solve for
the apparent mass of the plastic:
m'P
[17]
= m' – m'BW
When you do this calculation, mp’ will come out negative. This is expected and does not
mean anything spooky is happening, like anti-gravity. What is the reason that you expect this
behavior?
Once you have found the apparent mass of the plastic, you can use an expression analogous
to equation [15] to find the density:
[18]
ρ
(
= ρf
P
mP
m P – m'P
)
C. Measurement of Density of a Fluid Using Archimedes’ Principle
Use the brass block to determine the density of Wesson Oil, and predict whether the oil will
float on water or sink to the bottom. Since you already know the density of the brass block, by
using the mo you already measured and by measuring m’o in Wesson Oil, you can rearrange
equation [15] to calculate the density of the new fluid, Wesson Oil. Will Wesson Oil sink or float
on water?
[19]
ρ
f
=
ρ
o
(
mo – m'o
mo
)
-9 D. Please clean the beakers thoroughly at the end of the lab.
Equipment:
On Tables
eyedropper or pipet
Wesson Oil
Paper Towels
physics string (1 roll/table)
Wash bottles with distilled water (2/table)
In plastic boxes
100 ml grad cylinder
200 ml tall form beaker
400 ml beaker
balances, power supply, black plastic cover, Archimedes wire support
5 masses: sq. brass, cyl. brass, cyl. Al, cyl. plastic, irregular metal
Scissors
ruler
Report:
Introduction: Write a brief introduction concisely stating the purpose of the experiment, and a
concise summary of the methods that will be used.
Experimental: Describe the experimental apparatus and precisely what variables will be measured
and how they will be measured.
Results: Summarize the results of the experiment. Show sample calculations. If you are attaching
computer generated tables or graphs, briefly explain them here.
Discussion: Explain the significance of your results and their connection with more general
physical principles. Where it is possible, compare your numbers with accepted values. Explain any
sources of error.