A platform is placed on top of an air-filled bellows that is about the size of a bathroom scale (say, 1ft x 1ft x a few inches tall). A small (~1in) tube connects to the bellows at one side, and rises up vertically. The other end of the tube is open to the atmosphere. A man of average height and weight arrives. He notices that the open end of the tube is about at eye level. The man steps onto the platform and the bellows completely collapses since the air from inside is expelled through the tube, which is the only path for it to escape. After the man steps off the platform the bellows refills with air, by virtue of the tube.

The bellows is then filled with water (all air is removed). The man again steps onto the platform. How much water is expelled from the bellows?

The man’s identical twin comes along and joins his brother on the platform. How much more water is expelled?

At equilibrium, the pressures will equalize.  Since atmospheric pressure is equal at both ends, it can be ignored.  The pressure the man applies to the water is represented as:

[(mg)/A]

m=mass of man
g = acceleration of gravity (9.8 m/s²)
A = scale surface area

The pressure the water in the tube applies to the man is represented as:

ρgh

ρ = density of water (1000 kg/m³)
g = acceleration of gravity (9.8 m/s²)
h = height of open end of tube

Setting these equations equal and solving for h yields:

h = [m/(ρA)

Using these basic assumptions (m=100 kg, A = 0.1 m²):

h = 1 m

Doubling the mass of the man will double the height.

The volume of water expelled from the bellows is now determined by calculating the volume of 1 m of tubing.  Assuming a cross-sectional area of 0.0005 m², the volume of water expelled (into the tube) would be 0.0005 m³.  With a bellows height of approx. 0.1 m, the fraction of water expelled would be around 5%.  Note that this water would return to the bellows once the man stepped off.

If both men step on the scale, then the water column would rise to 2 m, and the fraction of water expelled would approach 10%.  However, the end of the tube should be below 2 meters since it is at eye level.  In that case, water would escape from the opening, thus allowing all the water to exit the bellows.  When the men eventually step off, the water contained in the tube will return to the bellows.

Posted by hoodat on 2007-03-01 14:21:35