BEAST PHYSICS TIPS
with Ashok Dibbawalla
Some Elementary Pressure Calculations Involving Water and County Executive Joel Giambra
What would happen if County Executive Joel Giambra were taken to the bottom of the ocean?
Dragging County Executive Giambra to the bottom of the ocean would serve as an excellent object lesson on the difference between compressible and incompressible substances.
To simplify the equations, let's start by placing County Executive Joel Giambra in a rigid vertical cylinder. If the cylinder has a diameter of ~44-1/2 inches, the ends of the cylinder will have an area of about 1 meter square. Let's imagine that the bottom of the cylinder is fixed to the walls, and that the top can move up and down in the cylinder like piston. Like a piston, the top also has a gasket that keeps the contents of the cylinder sealed inside. (See figure 1).
Imagine what would happen if we were to pour water into the top of the cylinder. The weight of the water would move the piston down until there was an upward force on the bottom of equal magnitude. Could County Executive Joel Giambra provide this force? Let's calculate its magnitude.
The product of the water's density and volume gives the weight of the water. If we were to fill the top to a height of 0.25 meters (~10 inches), County Executive Joel Giambra would have to support the weight of 0.25 cubic meters of water.
Water has a density of 1000 kg/cubic meter. This means that to hold up the cylinder, he would have to support 250 kg, or about 550 pounds. (See figure 2).
Now, of course he can't hold back the piston. But if this is true, one could wonder why divers aren't crushed after diving down only a few meters? The force that holds back the piston is provided by the air trapped underneath. As the piston moves down, the air pressure under the cylinder increases. The upward force it exerts on the bottom of the piston is given by the product of the air pressure and the area of the piston. As we fill the top, the sinking piston sinks compressing the air underneath until the pressure rises high enough to stop the downward movement. (This compression also increases the temperature in the cylinder, but by adding water slowly and letting the system cool off, we can ignore this effect.)
At 10 meters (~33 feet), the pressure is about double the atmospheric pressure at the surface. That means that the volume of the cylinder has been cut in half! (See figure 3).
Each additional 10 meters will raise the pressure by one atmosphere. As we journey with County Executive Joel Giambra deeper and deeper into the depths, we notice a pattern. The air around him, in his lungs and in his body cavities continues to shrink, but the fluids that make up his tissues continue to occupy the same volume! We call such fluids "incompressible."
After 1000 meters, the size of his fluids is the same, but the gases around him occupy less 1/100th of their former volume. Many of them have been dissolved into the liquids, like the carbon dioxide dissolved into a soft drink. By the time we reach the bottom of the Pacific Ocean at around 5,500 meters (~3.4 miles), we find County Executive Joel Giambra essentially liquefied. (See figure 4.)
All that remains is to calculate the volume of the bottom of the cylinder. It has been suggested that County Executive Joel Giambra weighs around 220 pounds, or about 100 kg. We'll estimate that the density of his remains is about the density of water. Dividing his mass by his density gives 0.1 cubic meters. The bottom of the cylinder is 1 square meter, therefore the height of the piston is 0.1 meters, or about four inches.