How Can Submarines Dive and Resurface? The Cartesian Diver Experiment

By Aline Pesch

I find it fascinating that ships float although they seem heavier than water. Even more fascinating is the science that explains how submarines dive and resurface. I discovered a YouTube video that shows an experiment in which a large plastic bottle is filled with water. A small glass test tube is filled with just enough water to keep it barely afloat in a bowl of water. A second test tube is kept empty, and both tubes are immersed with the open end pointing downwards in the large bottle, which is then capped. Squeezing the bottle, the tubes drift towards the bottom of the bottle and releasing the external pressure reverses the process.

Buoyancy moves a submerged object towards the surface of the surrounding liquid with a force that can be calculated as F_b = V_s ρ_L g, where F_b is the buoyant force expressed in N, V_s is the submerged volume in m³, ρ_L is the density of the surrounding liquid in kg/m³, and g is the force of gravity in N/kg. The variable external pressure produced by squeezing the plastic bottle forces water into the test tubes. The change in buoyancy per unit of change in hydrostatic pressure is greater in the test tube containing the larger volume of compressible air. It follows Pascal's Law: ΔP = ρg(Δh) where ΔP is the hydrostatic pressure in Pascal (SI unit) that describes the difference in pressure at two points in a fluid column that changes according to the weight of the fluid, ρ symbolizes the fluid density that is measured in kilograms per cubic meter (SI unit), g stands for the acceleration due to Earth's gravity measured in meters per second squared (SI unit) and Δh refers to the height of fluid above the point of measurement - or, in this experiment, the difference in elevation between the two points within the fluid column in meters (SI unit). As the water pressure increases, water enters the test tubes and compresses the contained gas, thereby decreasing the buoyancy of the test tubes. The process reverses when the gas expands again when the external pressure diminishes. Consequently, buoyancy increases and the test tubes float towards the top of the bottle again. The speed of displacement follows the change in ΔP and is influenced by the different volume of gas contained in each test tube.

While compressible, water is much less compressible than air. As predicted by the perfect gas law p = ρRT, increasing the pressure by the factor 2 will decrease the gas volume by 50%, i.e. its density doubles. Assuming that the volume of the empty test tube is 20 mL and that the second test tube contains 10 mL of air, the differential volume change is greater in the air filled tube, hence its buoyancy decreases at a higher rate than in the second tube. As shown in the video, the air filled tube displaces faster than the comparison tube. 

http://www.youtube.com/watch?feature=player_embedded&v=S3xxs1vkGAY