Archimedes’ Principle
Experiment 5
Objective: To illustrate the application of Archimedes’ Principle to the determination of
density.
DISCUSSION:
Archimedes’ principle states that a body immersed in a fluid experiences an
upward force due to the surrounding fluid and that this force is equal to the weight of the
fluid displaced by the body. We can make use of this principle to determine the density
of various substances. Density is defined as mass per unit volume, or in equation form:
m
  , where  is the density, m is the mass, and V is the volume
V
Consider a body immersed in water, as in Figure 1,
with a density greater than that of water. To keep the object
from sinking, we support it with a string. The tension, T, in
the string plus the buoyant force, B, of the surrounding water
is just sufficient to balance the weight, W, of the object. That
is,
W T B
(1)
T
B
Body
W
Figure 1: The forces on an
immersed body.
We can use this equation to find the buoyant force by weighing the body in water.
To find T, provided we suspend the body from the weighing scales using the string. The
buoyant force is equal to the weight of the displaced water. The volume of the displaced
water is the same as the volume of the body that is submerged. Therefore the ratio W / B
is the weight of the object to the weight of an equal volume, V of water. Keeping in mind
that weight is W=mg, dividing both numerator and denominator by gV, where g is the
acceleration due to gravity and the volumes are equal, we obtain the ratio of density of
the object to the density of water. That is,
 mb   mb g   W 

 
 

 b  Vb   Vb g   gVb  W




 w  mw   mw g   B  B

 
 

 Vw   Vw g   gVw 
Tupper
Body
(2)
Here b and w are the densities of the body of water
respectively. By definition, the density of water is exactly
1000 kg/m3, or 1 g/cm3and so Equation 2 can be used to
calculate the density of the body.
Tlower
Sinker
Figure 2: A floating body setup
with only sinker immersed.
5-1
If the body ordinarily floats in water, the buoyant force equals its weight. To be
able to use Equation 2 to determine the density, it is necessary that the body be
completely submerged in the water. We do so by attaching a sinker, as seen in Figure 2.
A tension Tlower in the lower string, along with the buoyant force on the sinker, supports
the sinker against the force of gravity. Note that this tension Tlower is the same whether the
body to be measured is covered with water or not, as long as the sinker is immersed. This
provides a straightforward way to measure the buoyant force B on the body.
First consider the beaker partly filled as shown in Figure 2. The water covers the
sinker but not the body. The only upward force on the body is Tupper. There are two
forces downward: Tlower and the weight Wb. Therefore
T upper,bodynotsubmerged  T lower  W b
(3)
If more liquid is added so that the body is also covered (as
shown in Figure 3), the liquid exerts an upward buoyant
force on the body. The net force is still zero, so
T upper  B  T lower  W b
or
(4)
T upper,bodyimmersed  W b  B  Tlower (5)
Tupper
Bb
Body
Wb
Bs
Tlower
Sinker
Ws
Figure 3: A floating body setup
entirely submerged.
The difference in these two measurements of Tupper (Equations 5 and 3) is just B, the
buoyant force of water on the body.
EXERCISES:
1. By estimate, order the objects in increasing density.
2. Calculate the density of denser-than-water objects using volume.
a. Using calipers1 or a ruler, and the appropriate volume formulae1, determine the
volume of as many of the objects as possible.
b. Use the scales to directly determine the mass of the objects.
c. Now calculate the density of each object.
3. Calculate the density of denser-than-water objects using Archimedes' Principle.
a. With string, suspend each object from the hook on the scales. Record this mass in
your table.
1
See the Appendix for accepted values.
5-2
b. Now move the small shelf above the pan of the scales and place the beaker filled
with water on the shelf. Suspend each object from the hook such that they are
completely submersed and find the object's apparent mass in water.
c. Using Archimedes’ Principle and equation 2, calculate the density of each object.
d. Determine the percent difference between the accepted values1 and the densities
you found using Archimedes' Principle.
4. Determine the percent difference between your values using Archimedes' Principle
and those you found in Part 1.
5. Calculate the density of less-dense-than-water objects.
a. Suspend the less-dense-than-water objects, and from each hang a sinker.
b. Make two mass measurements: first with the water covering only the sinker, and
then with the water covering both the body and the sinker.
c. Find the density of each body.
d. Calculate the percent difference from the accepted values2.
6. Comment on how well or how poorly you estimated the densities and why.
2
See the Appendix for accepted values.
5-3
Data Sheet – Experiment 5
Part 1
Estimate of objects by density (greatest to least):
Part 2
Object
Volume*
V
(cm3)
Mass
m
(grams)
Density
d=m/V
(grams/cm3)
* see appendix for appropriate formula.
Part 3
Object
Part 4
Object
Weight
of
both
W
(grams)
Weight in
air
W
(grams)
Weight
w/both
submerged
Tu, wet
(grams)
Weight in
water
T
(grams)
Buoyancy
B=W-T
(grams)
Density
W w
d
B
(grams/cm3)
Buoyancy
of both
Buoyancy
of sinker**
Buoyancy
of Body
B=W-Tu,wet
(grams)
Bs
(grams)
Bbody=B-Bs
(grams)
Density
Wbody  w
d
Bbody
(grams/cm3)
**As determined in part 3.
5-4