AMATH/ATMOS 505/OCEAN 511
Lab 2:
Buoyancy
Equipment: (purchased as of Oct 3, 2008)
• 4 ring stands with clamps (may need an additional clamp)
• 4 30mm-diameter, 1m-long, 2mm-thick glass tubes
• 4 #5 12 stoppers
• 4 17.5 mm test tubes
• balloons or rubber membranes
• rubber bands
• water
• salt
• plastic wrap to help invert test tubes without losing water
• permanent marker
Directions: Set up the cartesian divers ahead of time, to conceal the contents of the second
tube. Divide the class into several groups to think about these questions separately.
Goal: To develop intuition for how fluids at rest behave under gravity, pressure and buoyancy forces.
Useful information:
p = po − gρw (z − z0 )
FB = gρw V
Hydrostatic Balance
(Upward) Buoyant Force
where p is pressure, a force per unit area (Pa=N/m2 ), z is the vertical coordinate (m), p0 is the
pressure at z0 , g is acceleration due to gravity (9.8 m/ss ), ρw is density of water(kg/m3 ), FB is the
buoyant force (N), V is the volume of water displaced (m3 ), and Wdiver is the weight of the diver
(N).
AMATH/ATMOS 505/OCEAN 511
2.1
Cartesian diver 1
Setup directions: Fill the long tube with water. Fill the test tube partially with water, then
turn it upside down without spilling the water out of the test tube and put it in the long tube
so it floats just above the top. Experiment a few times to get the right amount of air into the
test tube. If there is too much air, the diver will not sink, if there is not enough air the diver
will not come back up. Make sure the water in the large tube comes all the way to the top. Cut
off the small end of the balloon and secure it to the top of the large tube with rubber bands.
The seal between the long tube and the balloon should be air tight and not have an air bubble
at the top. When you press on the balloon, the air in the small test tube contracts, and the
test tube should sink to the bottom. When you stop pressing, the air expands and the test tube
should rise to the top again.
Tips for setting up divers:
• Before placing the test tube in the long tube, mark the water level on the test tube using
a waterproof marker, which is useful for the next person to perform this demo as well
as making it easier to get the right water level in the other test tubes by comparing with
this test tube. If you make the wrong mark, you can use rubbing alcohol to remove most
permanent marker marks.
• Placing the test tubes into the long tube without spilling any water can be difficult. Wrapping plastic wrap around the end can help. Use a large enough piece of plastic wrap so
that you can pull out the wrap after inserting the test tube (the plastic wrap should not
be left inside the long tube).
(1) Why does the diver go up and down?
Changes in pressure increase/decrease the volume of the air bubble so that the diver alternately
becomes more or less dense than the surrounding water.
(2) Conceptually, what is the limit on the length of the large glass tube so that we can still have
the diver travel down to the bottom and back up?
When the tube is too long, the increase in hydrostatic pressure with depth becomes larger than
the change in pressure from the stopper, and the diver will be lost on the bottom with too small
an air bubble for recovery.
(3) Estimate the critical depth from which the diver will be unable to reascend to the surface.
AMATH/ATMOS 505/OCEAN 511
Strategy: Measure the initial volume v and the change in volume ∆v required to just make
the diver neutrally buoyant (hover completely submerged) when it is just below the surface.
The volume required to make the diver neutrally buoyant is independent of depth, since the
buoyant force needed to balance the weight of the diver is constant with depth. We wish to find
the deepest level at which the diver can be neutrally buoyant, which is the level at which the
hydrostatic pressure (without any pressure on the membrane beyond atmospheric pressure) is
great enough to give us the volume required for neutral buoyancy.
Compute ∆p due to the pressure exerted on the membrane, then determine the height of the
water column that will produce the same ∆p hydrostatically.
Using the ideal gas law and noting that the temperature in the air bubble does not change, the
∆p from the pressure exerted on the membrane is
pa v = nRT = (pa + ∆p)(v − ∆v)
pa ∆v
∆p =
v − ∆v
Using the hydrostatic balance, the equivalent pressure exerted by the water column at a depth
h is
∆p = ρw gh
pa ∆v
h=
(v − ∆v)ρw g
Assuming standard atmospheric values, an initial diver volume of 6 cm3 and a ∆v of 1 cm3 , we
find that if the diver drops below approximately 2.1m, it will be unable to return to the surface.
(Some divers may have had more initial buoyancy than others.)
Note that this approach does not require any assumptions about the mass of the diver, which
would be required by a calculation from an equilibrium between the buoyant force and the diver’s
weight.
2.2
Cartesian diver 2
Equipment: Same as above, plus salt.
This time prepare the long tube 1/3 to 1/2 the way from the bottom with a saltwater solution,
then carefully fill the remainder with freshwater. Don’t reveal the contents of the tube
to the class in advance!
(4) Why doesn’t the diver go to the bottom in this case?
The water in the tube is stably stratified with salty water on the bottom and fresh water on the
top. Here, we must consider the fresh water within the diver. When the diver hits the interface,
its is less than the salty water and it cannot sink further down.
(5) Experiment with different pressure changes (by pressing harder or softer on the balloon). How
does this change the motion of the test tube diver?
With a higher pressure change, the test tube will gain more momentum on its drop and so will
oscillate about the interface. Unfortunately to see this clearly, you need a really sharp interface
between the salty and fresh water. Mixing of the two fluids makes a sharp interface hard to
maintain.