Flotation and Archimedes' Principle

 "Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced." On floating bodies I, prop 5.

 The Greek mathematician Archimedes, is credited with the discovery of buoyancy. The story behind the discovery goes something like this: A king wanted a new crown and commissioned a snappy model from a prre-comuter version of crown-makers.com. Subsequently, the crown was fabricated and presented to his highness. The king wondered if he had a golden crown as ordered, or had the crown-maker defrauded the king by using heavy but inexpensive base metals (lead comes to mind) in making the royal hat. The problem was turned uver to Archimedes, a great mind of that, or of any, time. Ostensibly, while bathing at the public bath, the answer , and a very non-intrusive one at that, came to mib as he lowered his body into the tub. It is said that he ran through the streets naked saying over and again "Eureka" [I have found it..] The method was simple. Determine the mass of the crown by weighing it. Determine the volume of the crown by water displacement. Compare the density of the crown to the density of a gold bar

Consider a quantity of water, say a bathtub full. Any cubic centimeter of water is like any other cubic centimeter, each containing one gram of water. Let us isolate one cubic centimeter for observation purposes. Gravity acts on it like it acts on all the other cubic centimeters in its neighborhood. They are all interchangeable. Now what if we substituted for our water cube a solid cube of the same mass and volume. Gravity would treat this solid cube the same as its water brethren The solid would have neutral buoyancy and would stay wherever it was placed in the bath tub. Now let the solid have a slightly greater mass than its neighbors. Gravity exerts on this cubic centimeter a force greater that that applied to its neighbors; the solid sinks.

Now let's cause our substituting solid to have a mass less than the mass of its neighbors. Because the cubic centimeter of water above the solid is more massive, it weighs more and is pulled down to the lower position; the solid moves up one space. If this happens repeatedly, the solid winds up on top of the water surface; it floats. According to Archimedes, (see the proposition in the box above) a solid, now floating object will displace its own weight of the fluid it displaces. Say we have a solid whose mass is .5 g and whose volume is 1 cm^3. The solid will displace.5 g of water which in turn has a volume of .5 cm^3. Thus the floating object sinks until it has an apparent density of 1 g/cm^3.

 We can use the fact that the amount of the object sinks in a liquid depends on the relative density of the solid to the liquid to develop a device to measure the density of the liquid. It's called a hydrometer; see the image at left. The rod is bottom-weighted so that it floats in a vertical orientation. If it were completely submerged, its density would be that of the liquid; floating on top, its density would be zero. In fact, the rod floats partially submerged; the operator of the hydrometer can use the scale on the side of the hydrometer to learn the density of the liquid. A small toy known as a Cartesian diver relies on change in density to perform its trick. Fill a 2-two-liter soda bottle with water. partially fill a dropper with water and place it in the bottle; replace the cap tightly. Squeeze the bottle firmly and watch the diver go down; remove the pressure, the diver rises. It's a density thing. Check out this applet http://lectureonline.cl.msu.edu/~mmp/applist/f/f.htm The Galilean thermometer purports to measure temperature to +/- 2 oF. The floating weights have increasing densities top to bottom. When the room is cold, all weights are floating and they maintain their place in line because there is not enough clearance for a lower weight to pass the weight above. As the room warms, so does the liquid, lowering its density. If the room continues to warm, the density of the liquid becomes less than that of the lowest weight which then sinks to the bottom.. Each floating weight carries a metal tag indicating a temperature. The room temperature = the fluid temperature = the reading on the metal tag attached to the lowest floating weight

Buoyancy

We have seen what happens when a solid is placed in a fluid more dense than the solid; the solid floats. But fluids also have a bearing on lighter (actually: less dense) objects that are immersed in them also. Have you ever swum under water and attempted to move a rock out of its submerged position and onto the shore. The rock moves relatively easily while submerged but becomes a significant burden (like a rock) once it leaves the water.

Let us idealize the shape of our rock into a cube one foot on edge and submerge it in water so the the top surface is x feet below the surface. That means that the bottom surface of theof the block is x + 1 ft below the surface. Because the pressure at any given depth in a liquid is the same in any direction, an upward force called buoyancy makes itself known.

Buoyancy is easy to notice in water. The cradle that landed with the astronauts who landed on the moon was designed to support the weight of the lunar excursion module [LEM] while it was on the moon. The same craft that cradled LEM while on the moon would have been crushed by the the Earth's gravity because here g = 9.8 m/s/s rather than the paltry 1.6 m/s/s on the lunar surface. NASA scientists immersred the LEM craft in water to lower its weight so that it could be supported by flimsy lunar legs.