Recall:

• Density = r = mass/volume = M/V, and is measured in kg/m³ in SI units.
• Pressure = P = force/area = F/A, and is measured in N/m² = Pa.

Variation of Pressure with Depth

"If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium." The situations we will consider involve fluids in static equilibrium, at least at the initial and final instances. To be in equilibrium, "all points at the same depth must be at the same pressure", otherwise parts of the fluid would feel a net force, and would tend to move, contrary to the assumption of static equilibrium.

Note: Although individual molecules are free to move about in a fluid, this is not contrary to static equilibrium. The individual molecules move independently, and more or less randomly. On average, for every molecule that moves to the left, another moves to the right, and the net sum of all the motions is zero. The fluid motion mentioned above is a bulk motion, or a flow, where an entire collection of molecules has some net motion.

To understand how pressure varies with depth, let's imagine a container of fluid, and consider the forces acting on a column of fluid with area A, and extending from the top to a depth h. In equilibrium, there is no net horizontal force on this column. The force of air pressure acts on the fluid, exerting a downward force of F₀ = P₀A. The weight of the fluid also exerts a downward force of F_w = Mg, where M = rV = rhA. These downward forces must be balanced by an upward force. This upward force comes from the pressure of the fluid at depth h, so calling this pressure P, the upward force on the column of fluid is F_up = PA.

These are all the forces, and they must sum to zero, otherwise the system is not in equilibrium. PA - Mg - P₀A = 0,
or, inserting the expression for M in terms of r, and canceling the common factor of A,
P = P₀ + rgh.
Normally, P₀ stands for the atmospheric pressure, which has a standard (average) value of 1.01x10⁵Pa. As one goes deeper into a fluid, the pressure increases as rgh.

Notice that if P₀ is equal to atmospheric pressure plus some additional pressure, then the additional pressure shows up everywhere in the fluid. That is, the above expression for P depends on P₀ and the depth only. This fact is known as Pascal's principle:
Pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the containing vessel.
Applications of this principle include hydraulic systems like brakes, and garage lifts.

Pressure Measurements

A manometer is a simple device for measuring pressure. It consists of a tube containing a fluid, one end of which is open to the atmosphere, while the other end is sealed a fixed volume of trapped gas. (Drawing) Points A and B are at the same "depth" relative to the top of the fluid at the open end of the container. Therefore, the pressure at A will equal the pressure at B, and the pressure at B is P = P₀ + rgh, where h is the distance to the top of the fluid.

P is called the absolute pressure (meaning relative to absolute zero pressure, which is a perfect vacuum), and P - P₀ is called gauge pressure (meaning relative to atmospheric pressure, as is measured by many gauges). The manometer measures gauge pressure, since P - P₀ = rgh.

A barometer is another simple device for measuring pressure. A barometer is a long (>0.76 m) tube, filled with mercury (traditionally), then inverted into a dish of mercury, such that there is no gas inside the tube. The mercury will pull away from the closed end of the tube, leaving a vacuum -- P = 0 at the top of the mercury in the tube. Using our equation for P versus depth, putting a negative sign in front of h since it is higher, not lower than the open surface of the mercury, we have 0 = P₀ - rgh, or P₀ = rgh. (It is traditional to keep h a positive number.) So the barometer measures absolute atmospheric pressure.

At a standard one atmosphere of pressure, the height of mercury is 0.76 m = 760mm = 760 torr. The density of mercury is 13.595x10³ kg/m³ which means that standard atmospheric pressure is
P_0,standard = rgh = (13.595x10³ kg/m³)(9.8m/s²)(0.76 m) = 1.013x10⁵Pa.

Bouyant Forces and Archimedes' Principle

"Any body completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the body."

This upward force is called the bouyant force. The bouyant force is equal to the weight of the displaced fluid, and acts at the center of gravity of the displaced fluid. An object dropped into water is acted on by two net forces, its weight acting downward and the bouyant force acting upward. If the weight exceeds the bouyant force, the object sinks; if not, it floats. Since the weight is given by the average density of the object, r_avg, times the volume and g,r_avgVg, and the bouyant force is given by r_waterVg, we see that the object sinks if its density is greater than the density of water.

So how do fish and submersibles remain in equilibrium at fixed depths in water? They are each able to adjust their density to match the surrounding water density. By increasing or decreasing their density, they are able to change their depth up or down.

Example: P9.26

An object weighing 300 N in air is immersed in water after being tied to a string connected to a spring scale. The scale now reads 265 N. Immersed in oil, the object weighs 275 N. Find (a) the density of the object and (b) the density of the oil.

We will use P = P₀ + rgh. Since the information supplied in the problem is not pressure but force, we will multiply the above equation by A, and use F = PA to get:
F = F₀ + r_fgV
F is the force measured on the scale. The subscript f on r is to remind us that we use the density of the fluid the object is immersed in.

(a) To determine the density of the object, we need its mass and volume. The mass we get by dividing the weight (in air) by g. The volume we get by considering the volume of water displaced to cause the change in the scale reading. That is, V = (F - F₀)/r_waterg = (300 - 265 N)/(1.0x10³ kg/m³)(9.8 m/s²) = 3.6x10^-3 m³. The density is then r = M/V = F/gV = (300 N)/(9.8m/s²)(3.6x10^-3 m³) = 8.6x10³ kg/m³.

(b) Now that we know the volume of oil the object will displace, we can determine the density of the oil.
r_oil = (F - F₀)/gV = (300 - 275 N)/(9.8 m/s²)(3.6x10^-3 m³) = 0.71x10³ kh/m³.

Chapter 10: Thermal Physics

Temperature and the Zeroth law of Thermodynamics

We all know that a glass of ice water (ice and water) will eventually warm to room temperature if left out, melting the ice in the process. And a cup of hot tea will eventually cool to room temperature. But what is this thing we call "temperature"? I'm sure everyone has a general idea of temperature, but for physics we need a precise definition.

Consider a cooler in which we place ice and a warm can of soda. The warm soda will exchange energy (in the form of heat) with the ice, eventually cooling to the same temperature as the ice. We haven't defined what heat is. For now let's just say that heat is a form of energy which can be transmitted between things. Because the soda can exchange heat with the ice, we say that they are in thermal contact. When they stop exchanging heat energy, we say that they are in thermal equilibrium.

So what if we have 2 cans of soda, and we want to know if they are in thermal equilibrium. How can we tell? We would take a thermometer (a third object, C), and place it in thermal contact with the first can of soda (A). After the thermometer and soda have reached thermal equilibrium, we note the temperature. Then we place the thermometer in thermal contact with the second can of soda (B). Again, after reaching thermal equilibrium, we note the temperature. If the two temperatures are the same, then the two cans of soda are in thermal equilibrium.

This idea leads to the zeroth law of thermodynamics:
"If bodies A and B are separately in thermal equilibrium with a third body, C, then A and B will be in thermal equilibrium with each other if placed in thermal contact."

So finally, back to the initial question, what is temperature? "Temperature is the property that determines whether an object will be in thermal equilibrium with other objects. Two objects in thermal equilibrium with each other are at the same temperature."

Thermometers and Temperature Scales

Thermometers are devices for measuring temperature. The common thermometers consist of a quantity of mercury or alcohol (glycol) in a thin glass tube. As the temperature varies, the fluid expands and contracts, and its height in the tube changes. A scale is supplied to read off the height of the fluid directly in terms of temperature.

The simple thermometers described above have certain drawbacks. They don't operate below the freezing point of the liquid, and thermometers based on different liquids may not agree at all temperatures. Normally the thermometers are calibrated for 0°C (32°F) when ice and water are in equilibrium, and 100°C (212°F) when water and steam are in equilibrium.

There are three temperature scales you need to be familiar with, the Celsius, Farenheit, and Kelvin scales. You are probably familiar with the Celsius and Farenheit temperature scales. The Kelvin scale (the SI standard) measures temperature relative to absolute zero. Absolute zero is the lowest achievable temperature, corresponding to 0°K, or -273.15°C. One kelvin unit equals on celsius unit -- a change of temperature by 1°K equals a change of temperature by 1°C. The following table summarizes the different scales.

Thermal Expansion of Solids and Liquids

Common thermometers make use of the expansion of liquids with increasing temperature. This phenomenon is called thermal expansion.

On a microscopic scale, thermal expansion occurs because the average separation of atoms and molecules increases. The atoms and molecules in a substance are constantly vibrating, but on average they remain a certain distance apart. If the temperature of the substance increases, then the energy of the vibrations increases, meaning that the size (amplitude) of the vibrations increases. The result is that the average separation between atoms and molecules increases.

Empirically, we find that an object of initial length L₀ will undergo a change of length, DL for a temperature change DT, given by:
DL = aL₀DT.
The proportionality constant a is called the coefficient of linear expansion. The coefficents of linear expansion for a variety of substances near room temperature are listed in table 10.1 of the text.

Exapmple: P10.3

The New River Gorge bridge in West Virginia is a 518 m long steel arch. How much will its length change between temperature extremes -20°C and 35°C?

From table 10.1 we find the coefficent of linear expansion for steel to be a = 11x10^-6/°C. If L₀ = 518 m, and DT = 55°C, then
DL = aL₀DT = (11x10^-6/°C)(518 m)(55°C) = 0.31 m = 31 cm.