Message in the bottle
doc. dr. Gorazd Planinšič
Faculty for Mathematics and Physics, University of Ljubljana
and The House of Experiments, Ljubljana
I have spent last weekend with my family in Savudrija, small village on the northern Adriatic coast in
Croatia. September is perfect month to visit this place. Sea is still enough warm for swimming and there
are only few people on the beach. An ideal conditions for outdoor science activities without attracting
unwanted attention.
There are number of physics phenomena at the beach that offer the opportunity for exciting
experiments. This time I was looking for an experiment from the field of fluid static. I started with a simple
problem: how to measure the depth to which I can dive? The obvious answer is “by using a rope with a
stone tied to its end”. But there is not much physics in this approach and besides, this method is useful
only for limited depths and in very calm sea. Next, I remembered the problem about diving with a rubber
balloon. Assuming the water temperature does not change much at the depths that I can reach, I can
determine the depth from a change in balloon’s volume using Boyle’s law. But there are problems with
this approach too. It would be very difficult to take measurements of the balloon’s volume down there at
the sea bottom and by bringing the balloon up to the surface the information about change in volume
would disappear. Could I somehow “lock” the volume of the air down there under the sea and than bring
it out to the beach where I can measure the change? Of course – close it into the bottle!
Here is how I did it.
I took an empty 0,5 litre soda bottle with a cap. I dived to the desired depth (the bottle may get flatten
a little bit), turned the closed bottle upside-down (with a cap toward the sea bottom) and opened the
cap. At this moment some amount of sea water quickly moved into the bottle compressing the air
that was trapped in it. If you keep the bottle in upside-down position no air can escape from it. Than I
carefully closed the bottle, swam up to the surface and gave the bottle to my ten-year daughter Tina
who measured the height of the water column (Figure 1). (It is good to open the upright bottle before
taking measurements, since the pressure inside the bottle is now greater than the atmospheric pressure
and so the bottle may get slightly deformed.) In the meantime I used the ‘rope and stone’ method to take
independent measurement of the same water depth. We repeated the measurement at five different
depths. Later we used the marked heights to measure the volumes of the water columns by using
a graduated cylinder. We also measured the total volume of the soda bottle. The measured data are
presented in the first column of the Table.
V (ml)
V0 / V
hcalc (m)
hrope (m) ± 0,05m
65
85
127
149
187
8,3
6,3
4,2
3,6
2,9
1,4
1,9
3,1
3,8
5,3
1,1
2,0
3,0
3,8
5,3
Tabela: Diving depth was determined in two ways: by calculating it from the amount of water in the soda bottle
and by measuring it using a ‘rope and stone’ method. The total volume of the soda bottle was V0= 539 ml.
After the physical exercise it is time to do some exercise in physics. We need to relate the change in air
volume to the corresponding sea depth. The pressure at the distance h under the sea surface is the sum
of atmospheric pressure and the hydrostatic pressure
where ρ is sea water density (1,028g/ml for north Adriatic sea) and p0 = 105 N/m2 is the atmospheric
pressure at the sea level. The initial volume of the air in the bottle measured at p0 is V0. At increased
pressure p the volume of the air in the bottle decreases to V0-V where V is the volume of the water in the
bottle. Assuming the constant temperature of the water and using the Boyle’s law
one can express the depth h in terms of V in the following way. From
one finds
Using this expression I obtained the values listed in the third column of the Table. Though I never
doubted that physics works, I admit that I felt proud when I was showing the final results to Tina. Pity that
summer has gone? No problem. Any indoor swimming pool is perfectly suitable for repeating the same
experiment.
Figure: Tina is marking the height of the water column in the soda bottle (yes, we have forgotten to take
a ruler to the beach).