ARCHIMEDES' PRINCIPLE
The principle states that a submerged body experiences a "buoyancy force" equal to the weight of the displaced fluid (volume of body times fluid density times "g"). You've all heard of how he discovered this in his bath-tub; we'll get the same result by setting the vertical forces on the body equal to zero.
SPECIFIC GRAVITY
The procedure consists of weighing the sample in air and then in water. The weights are then used, following Archimedes' Principle, to compute the specific gravity.
FORCE ON A GRAVITY DAM
Our simple analysis does not deal with seepage, overflow and various other failure modes. We'll consider two possibilities. The first is that the force from water above the dam will simply cause the dam to slide away downstream. The second failure mode is that the water pressure may simply tip the dam over. Although the details are left for homework, the lecture will start the analysis.
We start by applying the conditions for stable equilibrium. The sum of the horizontal forces is zero. These forces are due to water pressure as in a previous class and friction (proportional to the (in air) weight of the dam). The sum of the vertical forces is also zero. These forces include the weight of the dam and the corresponding reaction force (no special name). To prevent tipping the moments around any point in the dam must sum to zero.
MANOMETRIC MODEL OF VOLCANO ELEVATIONS
The "manometric model" uses an analogy with a U-tube to explain this pattern. One branch of the U-tube is made of a column of water, crustal rock and mantle rock reaching from the sea-floor down to the magma source region at the base of the plate. The other branch is filled with light magma extending from the same source region up to the summit of the volcano. The time factor enters because, to a first approximation, the lithosphere thickness ("z" in km) increases with age ("t" in my) according to z=9.4 t^.5. Correspondingly the water depth ("d" in km) increases as d=2.5+.35t^.5. Thus, on an older plate, the "lithospheric" branch of the U-tube is heavier than on a younger plate and the magma (in the other branch) can be "boosted" to a higher elevation.
Data support the model out to a plate age (when volcano was built) of around 20 million years. Lack of agreement at older ages is attributed to "thermal rejuvenation" of the plate as it slides over the hot-spot. According to this concept, the temperatures and thickness of a plate are "reset" to values corresponding to those of a plate only 20 to 30 million years old when the plate moves over a hot spot..
Island arc volcanoes apparently grow to more or less the same elevation all along the arc. This has been taken to support the manometric model with a roughly constant magma source depth (depth of top of subducted slab beneath the volcano).
In contrast, continental arc volcanoes apparently have the same relief above the surrounding countryside, not the same elevation. This pattern has been interpreted in terms of a manometric model in which a magma source at a constant depth below the mean terrain elevation is at the base of the U-tube. You might want to update the data and calculations and test these three models.
WATER PARCEL OSCILLATIONS IN THE OCEAN
Applying Newton's Second Law in the vertical direction leads to a familiar differential equation. The form of the equation is identical to the linearized differential equation for a pendulum except for the names of the constants and variables. The solution is obviously a sinusoidal oscillation. The angular frequency is called the "buoyancy frequency" or "Brunt-Vaisala frequency" and is equal to the square root of the product of "g" times the "stability", E. "E" is just the density gradient divided by the density at the equilibrium depth.
If the density gradient is large, then the oscillation will have a short period (high frequency) whereas for a low gradient the periods will be very long (infinite for zero gradient). This model is at the heart of understanding internal waves in the sea.
REFERENCES
Any physical oceanography text referred to before.
INTRODUCTION
First we'll review buoyancy and Archimedes' Principle. Then we'll look at some problems in fluid statics. These include determination of specific gravity, forces on a gravity dam, models for volcano elevations and vertical oscillations in the ocean.
Archimedes lived in Syracuse, Italy, until he was killed in the Second Punic War between Rome and Carthage. He was renowned for his mathematical discoveries and practical engineering accomplishments (cranes, catapults, solar furnaces).
Specific gravity is just the ratio of the weight of an object to the weight of an equal volume of water at some standard temperature. In the c.g.s. system of units the specific gravity (dimensionless) is almost equal numerically to the density of water expressed in grams per cubic centimeter. You've all used specific gravity as a guide to identification of minerals and rocks.
As the name implies gravity dams are held in place by their own weight. Throughout the world dams impound vast volumes of water. Unfortunately they occasionally fail leading to death and destruction. These tragic events happen even in New Hampshire. The 1996 Alton failure cost one life and caused $5,000,000 damage (Marcoux, 1998).
The peak elevations of basalt volcanoes in the ocean (mostly hot-spot volcanoes) increase as the plate age during volcano growth increases. In other words only small volcanoes are found on young parts of the lithosphere plates but very high ones such as Hawaii are found only on plates that were old and thick when the volcanoes grew. Hawaii for example is on the order of a million years old but the plate is Cretaceous in age. Why should this pattern of elevations occur?
In much of the ocean the water density increases with depth. This is primarily because of the increasing pressure of the overlying seawater. A little "parcel" of water displaced upward from its equilibrium level is, of course, denser than its new surroundings and will sink according to Archimedes' Principle. Overshooting its natural level, the parcel finds itself lighter than its surroundings and will thus rise, overshoot, sink, overshoot, rise, etc..
Marcoux, G. J., 1998, Testing the self potential method for detecting seepage paths through earth dams in New Hampshire, MS thesis, University of New Hampshire, Durham, 115 pp.