Mechanics of liquids and gases. The Atmosphere and Weather.

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MECHANICS OF LIQUIDS

LIQUID PRESSURE

Pressure. The concept of pressure is associated with the concept of a

load distributed over an area. A pile of sand would be an example of a

load distributed, or spread out, over an area. Pressure is defined as

the force per unit area (pounds per square inch, for example) in a

situation where you have a load distributed over an area.

Consider a pile containing 2000 pounds of sand. The 2000 pounds of

weight is not concentrated at one point on the ground but is

distributed over the area the sand is sitting on. At each point under

the pile of sand there is some force acting on the ground measured in,

say, lbs/in², due to the weight of the column of sand above it

(envision a vertical column of sand measuring 1 inch x 1 inch in cross

section extending from the ground to the top of the pile). Now if we

view the entire pile of sand as consisting of 1 inch x 1 inch columns

then the sum of all the downward acting forces from all the columns

total to 2000 pounds and the force at any point under the pile comes

from the weight of the column of sand above it. The force at a

particular point under the pile is the pressure at that point. Another

example of a load distributed over an area: the load on the bottom of

a container containing a liquid such as water. If there is 20 lbs of

water in the container and the container bottom has an area of 10 in²

there is a pressure of 2 lbs/in² at each point on the bottom due to the

weight of the column of water above it.

Envision a vertical column of liquid 1 inch x 1 inch in cross section.

The pressure at any point in the column is due to the weight of the

liquid above it. In general, the pressure, p, exerted by a liquid of

density, D, at depth h is given by

p = hD.

Liquids exert pressure in all directions. The pressure exerted by a

liquid is directly proportional to the depth of the liquid and to the

density. It is independent of the area or shape of the container; it

also is independent of direction.

The total force, F, exerted by a liquid against a surface is given by

F = AhD

where A is the area of the surface, h is the average depth of the surface

and D is the density of the liquid. For horizontal surfaces the depth is

uniform. For vertical surfaces the average depth is usually equal to half

the depth.

Instruments for measuring water pressure.

Open manometer

Bourdon pressure gauge

Water pressure considerations are important in:

- water systems which utilize elevated water tank reservoirs to

generate water pressure

- design of submerged objects such as house foundations, submarines,

ship hulls, diving suits, dams, canal locks, etc.

PRESSURE APPLIED TO LIQUIDS

Liquids transmit pressure. Because liquids are incompresssible (or nearly

so) any pressure that is applied to a confined liquid is transmitted in

all directions.

Pascal's Principle: Pressure applied anywhere on a confined fluid is

transmitted with undiminished force in every direction. The force thus

exerted by the confined fluid acts at right angles to every portion of

the surface of the container, and is equal upon equal areas.

Liquid pressure can be used to multiply force. Envision a small piston

with an interior cross-sectional area of 1 sq. inch connected by a tube

to a large piston with an interior cross-sectional area of 100 sq.

inches and the entire system filled with a liquid. A force of 1 lb on

the small piston will be transmitted through the liquid to create a

pressure of 1 lb per sq. inch over the entire 100 sq. inch area of the

large piston giving a total force of 100 lbs on the large piston. This

principle is used in the construction of such things as hydraulic

presses, hydraulic jacks, hydraulic brakes and hydraulic lifts found in

service stations. All are machines which multiply force in this way.

Mechanical advantage of a machine that multiplies force: If an applied

force of 1 lb generates a multiplied force of 5 lbs the mechanical

advantage of the machine is 5. In general, if an applied force of f

generates a multiplied force of F the mechanical advantage is F/f.

Mechanical advantage of a hydraulic press:

mechanical advantage = A/a

where A is the area of the large piston and a is the area of the small

piston.

or,

mechanical advantage = D²/d²

where D is the diameter of the large piston and d is the diameter of

the small piston.

ARCHIMEDES' PRINCIPLE

An object that is less dense than water floats. An object that is denser

than water will sink but will appear to lose a part of its weight when

submerged (because it is pushed up by displaced water). A body which

is of the same density as water will appear to be weightless when

submerged, neither sinking nor rising (it will appear to have lost all its

weight).

Buoyant force. The upward force which any liquid exerts upon a body

placed in it is called the buoyant force.

Archimedes' Principle: The buoyant force which a fluid exerts upon a

body that is placed in it is equal to the weight of the fluid the body

displaces.

Note that Archimedes' Principle is valid whether a body is totally

submersed or only partially submersed in the fluid.

If a body is denser than a fluid it will sink in the fluid but will appear

to lose an amount of weight equal to the buoyant force on the body. If

a body has the same density as a fluid it will remain in equilibrium,

neither rising nor sinking and will appear to have lost its own weight.

If a body is less dense than a fluid it will float. A floating body

displaces its own weight of liquid. The fractional part submerged is equal

to the ratio of the density of the body to the density of the liquid.

Buoyant force on a totally submerged body:

Buoyant force = weight of body in air - weight of body when submerged

SPECIFIC GRAVITY

Specific Gravity. The density of a solid or liquid divided by the density

of water is called its specific gravity.

To find the specific gravity of a substance we use the formula:

Sp. Gr. = weight of substance / weight of an equal volume of water

Methods for determining the specific gravity of a solid.

1. Solids denser than water.

Sp. Gr. = w/(w-w')

where: w is the weight in air and w' is the weight in water

2. Solids less dense than water.

In this case the solid floats. To find the weight of an equal

volume of water we force it to sink by tying a sinker to it. The

specific gravity is then given by:

Sp. Gr. = w/(w'-w'')

where w is the weight in air, w' is the combined weight of the

solid in air and the sinker in water, and w'' is the combined

weight of both the solid and the sinker in water.

Methods for determining the specific gravity of a liquid.

1. The bottle method. Weigh a small bottle when empty. Fill it

with water and weigh it again. Then empty the water and fill it

with the liquid of unknown specific gravity and weigh

it again. By subtracting off the weight of the empty bottle

from the two weighings the specific gravity is computed from:

Sp. Gr. = weight of liquid / weight of water

2. Loss-of-weight method (or bulb method). The denser a liquid is,

the greater is the buoyant force it can exert. We can find the

relative weights of two liquids by comparing their buoyant forces

upon the same solid. A glass bulb or platinum ball is usually

used. First we weigh the bulb in air, then weigh it in water,

and finally weigh it in the liquid of unknown specific gravity.

Sp. Gr. = buoyant force of liquid / buoyant force of water

Sp. Gr. = (w-w'')/(w-w')

where w is weight in air, w' is weight in water, and w'' is

weight in liquid.

3. The hydrometer method. The commercial hydrometer works

on the following idea: A wooden rod, loaded at one end so that it

will float vertically, will sink in water until the weight of the water

it displaces exactly equals its own weight. If it is placed in a

liquid of unknown specific gravity, it will sink until it displaces a

weight of the unknown liquid equal to its own weight. If the rod

is uniform, the densities of the liquids displaced will be

inversely proportional to the depths to which the rods sink.

Sp. Gr. = depth rod sinks in water / depth rod sinks in liquid

The commercial hydrometer has a scale graduated in such a way that

the specific gravity of the liquid in which it floats can be read

directly. It has a lower bulb filled with shot or mercury to act as

a weight and a second larger upper bulb to compensate for the lower

bulb.

Uses of specific gravity.

- identifying rocks and minerals

- judging purity of a liquid

- determining charge of a storage battery

- testing the concentration of acids

- measuring the amount of alcohol in various alcohol-water mixtures

- checking the specific gravity of milk

- estimating the freezing point of antifreeze-water mixtures

THE ATMOSPHERE

ATMOSPHERE

Atmosphere. The layer of gases surrounding the earth that we call air.

The atmosphere is estimated to be from 500 to 5000 miles thick. The

layer of air surrounding the earth is an example of a load distributed

over an area. It presents a load distributed over the surface of the

earth in the same way that a pile of gravel presents a load distributed

on the area over which it lies or water in a container presents a load

distributed over the bottom of the container.

Layers of the atmosphere.

Troposphere. Extends upward from the earth's surface to a height of

from 6 to 10 miles. Contains about 75% of the weight of the

atmosphere. The temperature of the air decreases as we get further

out, reaching -65^o F at the top of this layer. Nearly all our clouds

and storms are produced in the troposphere.

Stratosphere. Extends from about 10 miles to 20 miles above the earth.

Has an almost uniform temperature of about -80^o F. At the top of the

stratosphere the atmosphere loses its light-scattering ability.

Chemosphere. Extends from 20 miles to 50 miles above the earth. The

temperature of the chemosphere is not uniform. It rises from about

-65^o at the top of the stratosphere to about 0^o F at 30 miles. It

then drops down to about -120^o F at an altitude of 50 miles. At the

lower edge of the chemosphere most of the ultraviolet rays from the

sun are filtered out.

Ionosphere. Extends from 50 miles to about 250 miles above the earth.

The temperature increases rapidly up through the ionosphere. It is

here that the aurora borealis is observed. The ionosphere is

important for radio broadcasting. It reflects radio waves back to the

earth and makes long-range short-wave transmission possible.

Exosphere. Extends from 250 miles to the outer edge of the atmosphere.

There is very little air in the exosphere.

Weighing air. Air can be weighed. Weigh a bottle containing air, then

pump out the air and weigh the bottle again. The difference is the

weight of the air in the bottle.

Density of air. One liter of dry air at a temperature of 0^o C and a

pressure of 760 millimeters of mercury weighs 1.293 g. Air is commonly

used as the standard for determining the specific gravity of gases.

Torricelli experment. About the middle of the seventeenth century men

were trying to find out why no pumps would raise water more than 32 feet

in the pipes of the deep wells they were digging. Evangelista

Torricelli (1608-1647), an Italian physicist, knew that air had weight

and he suspected that it was the pressure of the surrounding air that

pushed the water up a pipe from which the air had been pumped out. If

this were so, mercury, which is 13.6 times as dense as water would be

pushed up only 1/13.6 times as high in an exhausted tube. Torricelli

took a glass tube about 3 feet long and, after closing one end, filled

the tube with mercury. Placing his finger over the open end he inverted

the tube and set it in a bowl of mercury. When he removed his finger

the mercury dropped away from the sealed end until its upper surface

came to rest about 30 inches above the level of the mercury in the bowl.

The mercury, in descending from the top of the tube, left a vacuum

behind it, and it seemed this vacuum was able to hold up a 30 inch

column of mercury. Torricelli believed, however, that the liquid was

supported not by some mysterious sucking action of the vacuum, as was

commonly thought, but by the air pressing on the mercury in the open

bowl. Torricelli's belief in the pressure of the atmosphere was

confirmed by Pascal. Pascal reasoned that if the mercury column in a

Torricellian tube was actually sustained by the pressure of the

atmosphere, the height of the column should be less at higher altitudes.

Pascal arranged to have a Torricellian apparatus carried to the top of a

3000 foot high mountain in central France. When the apparatus was

assembled at the top of the mountain, the mercury column was found to be

3 inches shorter than it was at the base of the mountain.

Dull, Metcalfe, Brooks. Modern Physics. p. 66, 67

Pressure of the atmosphere. The air at sea level exerts a pressure which

counterbalances a mercury column 76 cm high, which is about 30 inches.

This represents a pressure of 14.7 lbs/in² and is known as a "pressure

of one atmosphere".

Barometer. A device used to measure the pressure of the atmosphere.

Mercurial barometer. A device using a column of mercury to measure the

pressure of the atmosphere.

Aneroid barometer. A barometer which employs a partially evacuated metal

box instead of a liquid to measure atmospheric pressure.

Using a barometer to measure altitude. Barometers have long been

used to measure altitudes. For comparatively small elevations the

barometer falls 0.1 inch for every 90 feet of ascent. Above a few

hundred feet the fall is less regular. At the top of Mt. Whitney in

California, which is 14,495 feet above sea level, the barometer

reading is a little over half the reading at sea level. At the top Mt.

Everest in the Himalayas, which is 29,002 feet above sea level, the

reading is less than 10 inches.

Altimeter. An aneroid barometer graduated to read altitude directly.

MECHANICS OF GASES

The volume occupied by a particular weight of gas depends on both the

pressure it is under and the temperature. As a consequence physicists

have both a standard pressure and a standard temperature that they use

when measuring gases. The abbreviation "S. T. P." is used to indicate

standard temperature and pressure.

Standard pressure for measuring gas volumes. The pressure of a column of

mercury 760 mm high.

Standard temperature for measuring gas volumes. The temperature of

melting ice, which is 0^ø C.

Boyles' Law. The volume of a dry gas varies inversely with the pressure

exerted upon it, provided the temperature remains constant. In equation

form Boyles' Law can stated as

pV = p' V'

where p is the original pressure, V is the original volume, p' is the

new pressure, and V' is the new volume.

The density of a gas varies directly with the pressure exerted on the gas.

Buoyant force exerted by a gas. Just as water exerts a buoyant force on

submerged objects, air exerts a buoyant force on objects submerged in

it. Archimedes' principle applies to gases as well as liquids.

References

Dull, Metcalfe, Brooks. Modern Physics.

Freeman. Physics made Simple.