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Mechanics of liquids and gases. The Atmosphere and Weather.
MECHANICS OF LIQUIDS
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 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
Instruments for measuring water pressure.
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
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
mechanical advantage = D2/d2
where D is the diameter of the large piston and d is the diameter of
the small piston.
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
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
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. 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
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
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 -65o 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 -80o 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
-65o at the top of the stratosphere to about 0o F at 30 miles. It
then drops down to about -120o 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 0o 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 0ø 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.
Dull, Metcalfe, Brooks. Modern Physics.
Freeman. Physics made Simple.
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