Archimedes' Principle©98
Experiment 11
Objective: To apply 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.
Consider a body immersed in water, as shown in
Fig. 1. The density of the body is greater than that of
water, and so 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
T
B
Body
(1)
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.
W
Figure 1. Archimedes' principle.
The buoyant force is the weight of the displaced water. The volume of displaced
water is, of course, the same as that of the body. Hence W / B is the ratio of the weight of
the object to the weight of an equal volume V of water. Dividing both numerator and
denominator of the ratio by gV, where g is the acceleration of gravity, we obtain the ratio
of the density of the body to the density of water. That is,
W
ρ
W
gV W
b
(2)



B
ρ
B W T
w
gV
Here b and w are the densities of body and water
respectively.
By definition, the density of water is exactly 1000
kg/m , and so Eq. (2) can be used to calculate the density
of the bodies. If the body ordinarily floats in water, the
buoyant force equals its weight. To be able to use Eq. (2)
to determine density, it is necessary that the body be
completely submerged in the water. We do so by
attaching a sinker of known weight and density to the
body, as in Fig. 2. The weight of the sinker is Ws. The
3
Tupper
Body
Tlower
Sinker
Figure 2. Use of the sinker.
11-1
buoyant forces on the body and sinker are Bb and Bs respectively. The combined weight
W and the combined buoyant forces B are thus given by
(3)
W  W W
b
s
and
BB B
b
(4)
s
Equation 1 continues to give the relationship between W, B, and the tension T.
The buoyant force on the sinker is the weight of water displaced by the sinker. This is
the product of the volume Vs of the sinker, and its density s, and the acceleration of
gravity g. Thus we write:
V 
s
and
W
s
s g
Bs   w gV s
,
.
(5)
(6)
These equations can be combined to give the buoyant force Bb on the body. Then Eq. (2)
can be used for the body alone to find the density of the body.
This method can also be used to find the density of a liquid, say a solution of copper
sulfate, by weighing a body of known weight W and density b in the liquid. In this case,
Eq. 2 becomes
B
ρ ρ
(7)
l
b
W
where l is the density of the liquid and B is the buoyant force on the body immersed in
the liquid. (Care must be taken in this measurement that the body not react chemically
with, nor dissolve in, the liquid.)
EXERCISES:
1. Determine the density of several sinkable objects.
a. With string suspend and weigh in water the various denser-than-water specimens
whose weights in air have been measured.
b. Using Archimedes' principle, calculate the density of each specimen.
c. Determine the percent deviation your calculated densities make with the accepted
densities of the specimens.
2. Determine the density of several floatable objects.
a. Suspend and weigh in water the less-dense-than-water specimen.
b. Calculate the density of the specimen.
c. Determine the percent deviation your calculated density makes with the accepted
density of the specimen.
11-2
3. Determine the density of an unknown solution.
a. Using a body whose weight and density are known, measure the tension in the
string when this body is suspended in a liquid whose density is to be determined.
b. Calculate the density of the liquid.
11-3