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Mechanics of liquids and gases. The Atmosphere and Weather.




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 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

      or

            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 -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.





References

 Dull, Metcalfe, Brooks. Modern Physics.

 Freeman. Physics made Simple.



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