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[Phys-L] Re: Buoyancy question



Antti Savinainen is correct when he says, "Hence the boat could float
even if there was much less water than the weight of the boat would
take." If indeed the boat floats, he assumes this means the water
displaced is less than the weight of the boat. I think that depends on
the definition of "displaced" water.

Antti suggested a container slightly larger than the floating object
such that the displaced water resides in a rather thin layer between the
object and the walls of the container. He implied this would allow the
water to rise in height sufficiently to create high pressure on the
bottom of the object without displacing sufficient water to equal the
weight of the object.

First I will give the results of an actual experiment, then I will
discuss it.

I have a hollow aluminum cylinder that has a density near one. In 20
Celsius water it floats with its top even with the water level. In 30
Celsius water it sinks. This experiment is done at 20 Celsius so it
just floats.

The mass of the cylinder is 15.6 grams the diameter is 19 mm and the
volume is 15.6 ml. It fits nicely in a 50 ml graduated cylinder with
inside diameter of 22 mm. Note thay if it just barely floats, it is
excluding a volume of water exactly the same as its volume of 15.6 ml.

If I put 10 ml of water in the graduated cylinder, a weight of water
that is clearly less than the aluminum cylinder weighs, will the
cylinder float or sink? You have to do some algebra to predict this,
but let's just drop the aluminum cylinder in and see what happens.

It floats.

So... did the cylinder float by displacing less weight of water than the
cylinder's weight?

With the cylinder floating I note that the bottom of the cylinder is at
the 4 ml mark. It went 6 marks below the original 10-ml water level.
The water level rose to the 15.6 ml mark, a change of 15.6 marks. The
top of the aluminum cylinder is also at the 25.6 ml mark. So, here is
the question... if the cylinder is floating in water that has risen to
the 25.6 ml mark such that the top of the cylinder is also at the 25.6
mark and the bottom of the cylinder is at the 4 mark... how much water
is the cylinder displacing?

If the cylinder is just submerged, isn't it displacing its volume of
water? Yet, its volume is 15.6 ml and we know there aren't 15.6 ml of
water in there. There are just 10 ml in there. What's the definition
of displaced?

Another way to view this is conservation of energy. Begin with the
cylinder just touching the top of the water at the 10 ml mark. As the
cylinder is lowered into the water its gravitational potential energy
(GPE) falls. The water level rises and the GPE of the water rises. If
the object eventually floats, the GPE it lost is equal to the GPE the
water gained. MgH = -mgh in which let's say M and H pertain to the
object and m and h pertain to the water.

MgH for the aluminum cylinder is (15.6 g)(g)(-6 marks). mgh for the
water must be minus this. About 4 ml of water did not move; they still
reside under the cylinder. About 6 ml of water moved into an annular
region that extends up to the 25.6 mark, a change of about 15.6 marks.
Thus, mgh for the water is (6 g)(g)(15.6 marks).

Conservation of energy is met.

If we define the displaced water as about 6 ml, then the cylinder did
not displace its weight of water. If we define the displaced water as
15.6 ml because the barely floating cylinder (of volume 15.6 ml) is
excluding water from that volume, then the cylinder did displace its
weight of water. If we define the displaced water as the mass of the
water multiplied by how far it moved ( i.e. [m Delta h], such that the
"weight of the displaced water" is [m Delta h]g ) then the displaced
water is equal to the weight of the cylinder.

So... how do we define displaced water?

I think I have always thought of it as the excluded volume. But I also
think I have often thought of it in the manner of the GPE analysis.

By the way, for the particular dimensions of graduated cylinder and
aluminum cyclinder I described, the aluminum cylinder does not float if
we start with <= 6 ml water in the graduated cylinder. In this case
there truly isn't enough water present because the aluminum cylinder
hits the bottom of the graduated cylinder before it has "displaced"
enough water to float.


Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu