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LESSON 1 ( Physics 101 )

Physics is a major science, dealing with the systematic study of the basic properties of the universe, the forces they exert on one another, and the results produced by these forces. It is the remaining core of naturl philosophy and concerns itself with questions of what underlies the interactions of matter, energy, space and time, and even with what constitutes reality. It is not surprising that a field that has produced the theories of relativity and quantum mechanics and has drastically altered our concept of the universe has an aura of mystery --- of being remote from everyday experience and impossible to understand. Once one becomes accustomed to looking for an explanation of various phenomena in terms of underlying scientific principles, it is possible to see physics everywhere. The flight of birds, the operation of a microwave oven, the color of the sunset, and the pitch of one’s voice all have basic explanations in physics. Those explanations can be understood by anyone, not just professional scientists.

           Physics is closely related to the other natural sciences and, in a sense, encompasses them. Chemistry, for example deals with the interaction of atoms to form molecules. Much of modern geology is largely a study of the physics of the earth and is known as geophysics. Astronomy deals with the physics of the stars and outer space. Even living systems are made up of fundamental particles and, as studied in biophysics and biochemistry, they follow the same type of laws as the simpler particles traditionally studied by a physicist.

The emphasis on the interaction between particles in modern physics, known as the microscopic approach, must often be supplemented by a macroscopic approach that deals with larger elements or systems of particles. This macroscopic approach is indispensable to the application of physics to much of modern technology. Thermodynamics, a branch of physics developed in the 19^th century, deals with the elucidation and measurement of properties of a system as a whole and remains useful in other fields of physics; it also forms the basis of much of chemical and mechanical engineering. Such properties as the temperature, pressure and volume of a gas have no meaning for an individual atom or molecule; these thermodynamic concepts can only be applied directly to a very large system of such particles. A bridge exists, however, between the microscopic and macroscopic approach; another branch of physics; known as statistical mechanics, indicates how pressure and temperature can be related to the motion of atoms and molecules on a statistical basis.  

Physics emerged as a separate science only in the early 19^th century, until that time a physicist was often also a mathematician, philosopher, chemist, biologist, engineer, or even primarily a political leader or an artist. Today, the field has grown to such an extent that with few exceptions modern physicists have to limit their attention to one or two branches of the science. Once the fundamental aspects of a new field are discovered and understood, they become the domain of engineers and other applied scientist. The 19^th century discoveries in electricity and magnetism, for example, are now the concentrations of electrical and communication engineers; the properties of matter discovered at the beginning of the 20^th century have been applied in electronics; and the discoveries of nuclear physics, have passed into the hands of nuclear engineers for applications to peaceful or military uses.

MATHEMATICS as a language of science

     Mathematics is the language of physics; that is when ideas in science are expressed in mathematical terms:        

                   1. They are unambiguous.    

                   2. They do not have double meanings, that so often confuse the discussion of ideas expressed in

                        common language.

                   3. They are easier to verify or disprove by experiment.

                   4. The methods of mathematics and experimentation led to enormous success in science.

                   5. The abstract mathematics developed by mathematicians is often years later found to be the

                         exact language by which nature can be described.

    Mathematics is the language of physics does not mean that mathematics is physics or physics is mathematics.

THE SCIENTIFIC METHOD – is a method that is extremely effective in gaining, organizing, and applying new knowledge. The steps are :

1.      Recognize a problem.

2.      Make an educated guess --- a hypothesis. Hypothesis is an educate guess that is only considered factual after it has been demonstrated by experiments. If a hypothesis has been tested over and over again and has not been contradicted it may become known as a law or principle.  

3.      Predict the consequences of the hypothesis

4.      Perform experiments to test predictions.

5.      Formulate the simplest general rule that organizes the three main ingredients --- hypothesis, prediction, and experimental outcome --- into a theory.

The success of science has more to do with an attitude common to scientists than with a particular method. This attitude is one of inquiry, observation, experimentation and humility.

THE DOMAIN OF PHYSICS

A.  According to size of objects studied

     1.  Quantum domain – the domain of small objects. Objects are considered small if their sizes are

            comparable to or smaller than the size of an atom.

     2.    Non-quantum domain – the domain of large objects. Objects are considered large if they are larger 
             than the size of an atom.

B.  According to speed of objects studied

     1.  Relativistic domain – the domain at high speed, that is if the speed of the moving object is 
          comparable to the speed of light.

     2. Non-relativistic domain – the domain at low speed, that is the speed of the moving object is less than 
          the speed of light.

            

C.  Newtonian domain – a combination of the division according to size and speed. It is the domain of large

            objects at low speeds, the one we deal in our daily lives. (In honor of Sir Isaac Newton, the 17^th century physicist who played the key role in developing the physics of large objects moving at low speed).

            

D.  Mechanics – is the study of the relation between the force and the resulting motion. It seeks to account  

               quantitatively for the motion of objects having given properties in terms of the force acting on them.

      1.  Newtonian mechanics – is the mechanics of the Newtonian domain. It deals with systems containing 

            objects which are large and which move at low speed.

2. Relativistic mechanics – is the mechanics of the relativistic domain. In 1905, Einstein showed that a     different approach was necessary for the study of objects moving at speeds so high as to be                  comparable to the speed of light.

      3. Quantum mechanics – is the mechanics of the quantum domain. It was developed about the same time

           with relativistic mechanics by Max Planck, Louis de Broglie, Erwin Schrodinger and others. They 
           found out that the Newtonian mechanics could not explain the motion of objects whose size is in the 
           atomic scale or smaller.

 E.   Electromagnetism – is the study of the properties and consequences of the electromagnetic force, 
         which is one of the fundamental forces in nature. The fundamental forces are gravitational force, 
         electromagnetic force, strong nuclear force and weak nuclear force.

 F.   Solid-state physics is a branch of physics that deals with the properties of solids. A particular problem 

            in solid – state physics, for instance the properties of materials use in transistors, is solve by 

            employing the mechanics of whichever domain is most appropriate.  

 G.  Heat  and Thermodynamics

   THE FUNDAMENTAL MEASURABLE QUANTITIES IN PHYSICS

    1.  Length           3.  Time                   5.  Luminous intensity             7.  Molecular quantity

    2.  Mass             4.  Temperature        6.  Electric charge ( current )

   THE FUNDAMENTAL MEASURABLE QUNATITIES IN MECHANICS

               1.  Length                 2.  Mass                       3.  Time

  Measurement is a scientific comparison between an unknown quantity to a fixed known quantity called standard.

 Systems of Measurement

1.      English system  (British Engineering system) – originated in England

2.      Metric system – originated in France

  Systeme International d’Unites ( SI ) adopted by  the International Bureau of Weights and Measures in 
  1960.  The units of the MKS is adopted as the base units of the SI system.

Base Units of each System of measurement

Measurable Quantities in Mechanics	Metric System		English System
	CGS	MKS	FPS
Length	Centimeter ( cm )	Meter ( m )	Foot ( ft )
Mass	Gram ( g )	Kilogram (kg )	Slug ( lbm )
Time	Second ( s )	Second ( s )	Second ( s )

Reasons for adopting the Metric system:

1.      It is scientifically planned.

2.      It is a decimal system.

3.      It is universally accepted.

 DISADVANTAGES OF THE ENGLISH SYSTEM

            1 yard = ( King Henry I ) distance from the tip of his nose to the end of his thumb

            1 inch ( 1324 ) = length of three grains of barleycorns laid end to end

            1 mile = 1000 double step of an average soldier

            1 foot = length of the foot of the king

THE CONCEPT OF THE METER

                       To be discuss in class with demonstrations

MAKING AND RECORDING MEASUREMENTS

Reasons for uncertainty in Measurements

1.      The limitations inherent in the measuring instrument.

2.      The conditions under which the measurement was made.

3.      The different ways under which the person uses or reads the measuring instrument.

Terminologies:

1. Fundamental or base unit – the standard unit for length, mass and time.
2. Derived unit a combination of any of the three fundamental base units; i. e.  m/s, m/s²,  ft², m³ etc.
3. Accuracy – refers to the closeness of a measurement to the standard value for a specific physical quantity. It is express either as an absolute error or relative error.
4. Absolute error ( E_A ) is the actual difference between the observed ( O ) or measured value and the accepted value ( A ).

                                                      E_A = | O – A |

5. Relative error ( E_R ) I expressed as a percentage error and is often called a percentage error.

                                                      E_R = E_A/A

6. Absolute deviation ( DA ) is the difference between a single measured value ( O ) and the average ( M ) of several measurements made in the same way.

                                                                                                           

                                                      D_A = | O – M |

7. Relative deviation ( D_R ) is the percentage average deviation of a set of measurements.

                                                      D_R = D_A / M

8. Precision is the agreement among several measurements that have made in the same way. It tells how much reproducible the measurements are and is express in terms of the deviation.
9. Tolerance is the degree of precision obtainable in a measuring instrument.
10. Significant figure are those digits in a number that are known with certainty plus the digit that is uncertain.

HOW TO SUCCEED IN PHYSICS

LESSON 9 : FLUIDS and PRESSURE

FLUIDS AND PRESSURE

      Fluid is any substance that cannot maintain its own shape or has no rigidity. Fluid is a term applied to liquids and gases for they readily and they do not resist shearing stress. Fluids can flow and alter its shape to conform to the outlines of its container. Both liquids and gases have many characteristics in common. Liquids are nearly incompressible, while gases are easily compressed. Liquids tend to have greater density than gases. The gaseous phase of a substance usually exists at higher temperature, hence gas molecules are able to break free from one another.  Gases are able to escape from an open container, whereas liquids cannot.

PRESSURE

       Pressure is the force applied per unit area, P = F/A where F is the force applied and A is the area on which it is applied. Pressure is one of the important concepts in fluids. The definition is valid for all phases of matter: solid, liquid or gas. Examples of pressure: record player needle exerts pressure  on a disc record, water near the bottom of a pool exerts pressure on a swimmer’s eardrum, and atmospheric pressure changes weather conditions. Measurements of pressure are common : Tires must be inflated to correct pressure, blood pressure should stay within normal range, and too much pressure in the eye (glaucoma ) can cause blindness.

      The SI unit for pressure is Pascal ( Pa ); 1 Pa = 1 newton / m². An atmosphere ( atm ) is also a unit of pressure, where one atm = the average pressure due to the weight  of the atmosphere at sea level.

       Pressure is as important as the force creating it. If someone pokes you with his finger, you will certainly feel it. If, however, a nurse pokes you with a hypodermic needle using the same amount of force, you just don’t feel it – the needle breaks the skin. The same force applied to a smaller area creates a larger pressure and has a much different effect.

      Fluids as well as solids, can exert pressures. Consider the water in a square container. If the water has a mass of 10 kg, its weight of 98 N must be supported by the bottom of the container. If the bottom has  an area of 0.1 m², then the pressure due to  the weight of the water on the bottom is 980 N / m². This computation is valid only for containers with straight sides. Another example is atmospheric pressure, which is caused by the weight of air. Atmospheric pressure is 1.013 x 10⁵ Pa at sea level which means a column of air 1m on a side extending to the top of the atmosphere weighs 1.013 x 10⁵ N. Stationary fluids always exert forces perpendicular to surfaces whether that direction is up or down, left or right. The reason the force is always perpendicular to the surface is that fluids cannot withstand shearing or sideways forces and therefore cannot exert sideways forces.

CONVERSION FACTORS FOR VARIOUS UNITS OF PRESSURE

Conversion to N / m2 or Pa			Conversion to  atmosphere, atm
1.0  atm	=	1.013 x 105  N / m2	1.0  atm	=	1.013 x 105 N / m2
1.0 dyne / cm2	=	0.1 N / m2	1.0  atm	=	1.013 X 106 dyne/ cm2
1.0 kg / cm2	=	9.8 x 104 N / m2	1.0  atm	=	1.03 kg / cm2
1.0 lb / in2	=	6.9 x 103 N / m2	1.0  atm	=	14.7 lb / in2
1.0 mm Hg	=	133 N / m2	1.0  atm	=	760 mm Hg
1.0 cm Hg	=	1.33 x 103 N / m2	1.0  atm	=	76 cm Hg
1.0 cm water	=	98.1 N / m2	1.0  atm	=	1.03 x 103  cm water
1.0 bar	=	1.0 x 105 N / m2	1.0  atm	=	1.013 bar

Problems

1. A woman wearing high-heeled shoes places about 50 % of  its full weight  on the  single  heel when  walking. If the woman has a mass of 60 kg, determine the pressure on the ground under one heel if the area of contact is 2.25 cm². How would this pressure compare with the pressure exerted the elephant’s foot which is circular with diameter of 30 cm assuming that it is standing on four legs

 and  with mass of 3 650 kg.

2. Determine the pressure in N / m² and atm exerted by a phonograph needle on a record surface if the needle supports 2.25 g on a circular area 0.48 mm in diameter. 

3. Calculate the force exerted on one side of a 3 m by 5 m wall assuming normal atmospheric  

    conditions.

4. A hammer exerts a force of 40 N on a circular surface 2 cm in diameter. Determine the pressure.

BOYLE’S LAW

       Robert Boyle ( 1627 – 1691 ) advanced the study of gases using an air pump made by Robert Hooke. Boyle observed the relationship between the pressure and the volume of an enclosed gas at constant temperature. Boyle’s law states that at constant temperature the pressure exerted by a gas is inversely proportional to the volume in which it is enclosed.  

                                                P a 1/ V  ==>  PV = k  or PV = constant,

where P is the gas pressure, V is the volume  and the value of the constant depends on the initial conditions. A complete statement of Boyle’s law includes the condition that both the temperature and the amount of gas must be held constant. Alternatively, Boyle’s law can be written in the form  P₁V₁ = P₂V₂ ; where the subscripts 1 and 2 refer to the different physical states of the same sample of gas with the temperature held constant.

This simple dependence of pressure in a gas on the volume it occupies has many applications. If the volume of the gas is doubled, the pressure is reduced by half of its original value. An example is the air pressure in a tire, which results from putting a large volume of air into the smaller volume of the tire, thus making the pressure greater than the atmospheric pressure. Another example is that to drink through a straw, you expand the volume of the air in your mouth, dropping its pressure and allowing the larger atmospheric pressure to force the fluid up the straw.

 1. A cylinder with height of 20 cm and cross sectional area of 0.40 m² has a close-fitting piston that  may be moved to change the internal volume of the cylinder. Air at 1.013 x 10⁵ N/m² fills the cylinder. If the piston is pushed until it is within 8 cm from the end of the cylinder, what is the new pressure of  

the air? Assume that the temperature of the gas remains constant and that the volume of gas in the 

gauge is small compared with the volume of the cylinder.     

 2. A 150 cm³ gas initially at 500 kPa is allowed to expand until is pressure is reduced to 125 kPa. What  is the volume of the gas?

  

PASCAL’S PRINCIPLE

 One pioneer in the Physics of fluids was the French philosopher and scientist Blaise Pascal (1623 – 1662). He discovered an important property of stationary fluids: They can be used to transmit pressure to a place other than where the pressure is created and is not diminished while in transit.  

This is Pascal’s principle :

           Any pressure applied to a confined fluid will be transmitted undiminished to all parts of the fluid.

 This means that the applied pressure will be added to whatever pressures already exist in the fluid. One manifestation of Pascal’s principle is that the total pressure at the bottom of he lake is the sum of the pressure due to the weight of water plus the atmospheric pressure. According to Pascal’s principle, atmospheric pressure is transmitted undiminished to the bottom of the lake. Because water is not rigid, it cannot support the weight of the atmosphere without transmitting it to all parts of the lake. The effect of atmospheric pressure often cancels or is negligible, it is tiresome to always add atmospheric pressure to get total pressure. Gauge pressure is the pressure above or below atmospheric pressure. Total pressure or absolute pressure is gauge pressure plus atmospheric pressure.

                                                            P_t = P_g + P_atm  

                  where  P_t = total pressure, P_g = gauge pressure, P_atm = atmospheric pressure

An easy way to remember this equation is to recall that a tire gauge reads zero when the tire is flat, even though a flat tire with a large hole obviously contains air at atmospheric pressure. The flat tire has a gauge pressure of zero and a total or absolute pressure of 1 atm. We shall use the convention that all pressures are gauge pressures unless otherwise specified. Most pressure-measuring devices yield gauge pressures, and in most circumstances gauge pressure is of greater concern than total pressure. In the application of Pascal’s principle, it is important to note that applied pressure, not force, is transmitted undiminished to all parts of the fluids. Consider a hydraulic system consisting of two cylinders connected to one another by a tube and filled with a fluid as shown in the figure below.

 

The fluid used in the hydraulic system is incompressible. When a force  F₁ is exerted on the small piston, fluid flows from the small cylinder to the large cylinder, causing the large piston to move upward and exert an upward force to lift a load connected to it. Pascal’s principle states that the pressure created by exerting force F₁ on the small piston,  P₁ = F₁ / A₁, is transmitted undiminished to all parts of the fluid. Therefore

                                                P1 = P2   ==>   F₁ / A₁  =  F₂ / A₂

      Pascal’s principle holds for gases as well as for liquids, with some minor modifications due to the change in volume of a gas when the pressure is changed. In most hydraulic systems a small force is put into the system and causes a large force to emerge from the other end of the system. The small cylinder is called the master cylinder and the large cylinder the slave cylinder. For example, the car brake system have a master cylinder connected to four slave cylinders, one for each wheel. A small amount of force exerted by the driver is transformed into a larger amount of braking force exerted by the slave cylinder on the brake drum of each wheel. The frictional force exerted by the brakes on the brake drum causes the car to slow down speed or to stop.      

  1. A hydraulic jack lifts a 1,200 kg car by applying a force the small piston. If the diameters of the    

          large and small pistons are in the ratio 10 : 1, and the diameter of the large piston is 6 cm,  

          determine the force exerted by the operator on the small piston.

 2. The large piston of a hydraulic press  supports a dentist’s chair and the dentist wants to lift the      

patient by stepping on the pedal directly on top of the small piston. Find the force exerted by        the dentist if the patient plus chair have a mass of 125 kg and the large piston has radius of 6 cm and

 the small piston has radius of 1.2 cm.

     Hyperbaric medicine treats many physical problems through the application of high-pressure air or  air-oxygen mixtures. Hyperbaric oxygen therapy uses a special chamber, sometimes called a pressure chamber, to increase the amount of oxygen in the blood. Some, but not very many, hospitals have a hyperbaric chamber. Smaller units may be available in outpatient center. The air pressure inside a hyperbaric oxygen chamber is about two and a half times greater than the normal pressure in the atmosphere. By Pascal’s principle, the pressure is distributed throughout the hyperbaric chamber. This helps your blood carry more oxygen to organs and tissues in your body. The increased oxygen intake is useful in treating a variety of problems such as wounds, especially infected wounds, heal more quickly. The therapy may also be used to treat:

1.    Air or gas embolism

2.    Bone infections (osteomyelitis) that have not improved with other treatments.

3.    Burns

4.    Carbon monoxide poisoning

5.    Certain types of brain or sinus infections

6.    Decompression sickness (for example, a diving injury)

7.    Gas gangrene

8.    Necrotizing soft tissue infections

9.    Provide enough oxygen to the lung during a procedure called whole lung lavage, which is used to clean an entire lung in patients with certain medical conditions.

10.  Radiation injury (for example, damage from radiation therapy for cancer)

11.  Skin grafts

12.  Wounds that have not healed with other treatments (for example, it may be used to treat a foot ulcer in someone with diabetes or very bad circulation)

PRESSURE DUE TO WEIGHT OF A COLUMN OF FLUID

                                                P  =  F / A  =  mg / A

                                                but  m  =  rV  

                                                ==>  P  =  rVg / A    and  V = Ah 

                                                ==> P = r A h g / A 

                                                ==>  P = h r g

       The pressure due to the weight of a column of fluid depends only of the depth in the fluid and the density of the fluid. Pressure at the surface of a fluid is zero since h is zero. As it goes deeper the pressure increases in proportion to the increase in height.  

        Another manifestation of how pressure depends only on depth and density, is found in the intravenous ( IV ) administration of fluids. The pressure due to the IV fluid at the entrance of the needle is proportional to h, the height of the surface above the needle. Paths of the fluid has nothing to do with pressure but is dependent on h,  P = h r g. Applied pressure can be adjusted by raising or lowering the IV bottle relative to the patient.

Consider a situation where a medicine dropper is filled with fluid. First air is squeezed out of the bulb with the tip beneath the fluid surface as shown in Figure 1. The bulb is then released, and being elastic, returns to its original larger size. The volume of the gas in the bulb is then increased while the pressure decreases to a value less than the atmospheric pressure. The atmospheric pressure is then able to push fluid into the dropper at height h. This situation is analogous to how a hypodermic syringes are filled with fluid and how we drink soft drink from a straw. The height h to which the fluid rises in a medicine dropper is related to the pressure in the bulb, P_bulb , the lower the pressure in the bulb the higher the fluid rises.

                                                                              

                                  Figure 1   



                                                                           

                                 Figure 2

Consider the dropper in figure 2. The total pressure must be the same at both points mark X, for they are at the same vertical height. The total pressure at the outer X is atmospheric, P_atm. The total pressure

At the inner X is the pressure due to the weight of the fluid plus the pressure of the air in the bulb which is   h r g + P_bulb. The two pressures must be equal, so

                                                         P_atm =  h r g  +  P_bulb    solving for  h

                                                             h  = ( P_atm –  P_bulb) / r g

The pressure, P_bulb is an absolute pressure. All quantities in the expression is constant except the pressure in the bulb; the smaller this pressure, the higher h is. Since the pressure in the bulb cannot be less than zero, there is a limit to how high h can be.  For water, the maximum value of  h  is 10.3 m when P_bulb is zero. The larger the density of a fluid, the more difficult it is to raise h. The largest value of h obtainable for mercury is 0.76 m only.  The height to which fluid rises is directly related to the pressure in the bulb, hence the height of a column of fluid can be used to measure pressure. Many pressure-measuring devices are using this phenomena to measure pressure.

MEASUREMENT OF PRESSURE BASED ON PASCAL’S PRINCIPLE  AND  P = h r g.  

         Pascal’s principle states that any pressure applied to a confined fluid is transmitted undiminished to all parts of the fluid. Thus, fluid can be use to transmit pressure to a gauge at a convenient location. As an example, blood pressure can be measured without putting a gauge into the body. Furthermore, pressure is transmitted undiminished, so the measurement can be very accurate. In a blood pressure measurement, an inflatable cuff is placed on the upper arm, and inflated until blood flow is cut off in the brachial artery. Pressure is created by squeezing the bulb and is transmitted by air in the tubes (a confined fluid) to the cuff and to the gauge. The wall of the cuff transmits the pressure to the arm and through it to the artery. When the applied pressure exceeds the heart’s output pressure , the artery collapses. The person making the measurement slowly releases the air in the cuff, lowering its pressure, and listens for flow to resume when pressure in the cuff becomes lower than the maximum heart output. 

         Systolic blood pressure, ( when the heart is contracted ) the maximum blood pressure is recorded together with the lower pressure called the diastolic pressure (when the heart is relaxed between beats). Diastolic pressure is the minimum pressure the circulatory system experiences and can be detected as a change in the sound of the blood flow through the partially restricted artery. The main point here, based on Pascal’s principle, is that all these pressures are transmitted undiminished and the pressure read by the gauge is truly representative of the pressure in the heart.  It is important for the cuff  to be at the same level as the heart in a blood pressure measurement. Any effect  due to the weight  of the air in the cuff and connecting tubes is negligible because the density of air is very small. The gauge can be placed at any convenient location, such as on the wall, a table, the floor, or anywhere in between, but the cuff must be at the same level as the heart.

Manometers and analogous devices are in common used such that pressure units related to the use of these devices have been developed. A manometer is often a U-shaped tube filled with fluid usually water or mercury. A single-tube manometer filled with mercury is commonly used for measuring blood pressure. The pressure in the cuff is  h r g, where h is the height of the mercury and r  is the density of mercury. Common blood pressure measurements produces value of h  ranging from 8 – 300 mm, so h is commonly recorded in “millimeters of mercury” ( mm Hg ). The mm Hg is almost used universally to measure blood pressure instead of performing the product of  h r g for each individual measurement. A normal blood pressure of  100 over 80 means that the systolic pressure is  120 mm Hg and the diastolic pressure is  80 mm Hg.  If the fluid used in the manometer is water, h would also be measured in  centimeters of water ( cm H₂O ).

Mercury barometers measure atmospheric pressure in a manner similar to manometers. A tube closed at one end is filled with mercury and inverted with its open end in a dish of mercury. Extreme care must be considered in order not to allow air inside the tube. If the tube is long enough, part of the mercury will flow out of the tube thereby leaving a vacuum at the top. In a mercury barometer, the P_bulb is zero ( an absolute pressure of zero is a vacuum ). The pressure due to the weight of mercury must be equal to the atmospheric pressure.  Another way of interpreting this is to say that since above the mercury is vacuum, the atmosphere is able to raise the mercury to a height  h, where the pressure due to its weight,  P = h r g, equals atmospheric pressure. Atmospheric pressure varies with  altitude and weather conditions. The normal atmospheric pressure has a value of h equal to 760 mm Hg.     

Problems :

1.   A nurse administers medication in a saline solution to a patient by infusion into a vein in the

   patient’s arm. The density of the solution is 1000 kg/m³, and the gauge pressure inside the vein is

   2.7 x 10³  Pa. How high above the insertion point must the container be hung so that there is

   sufficient pressure to force the fluid into the patient ?

2.  A scuba diver searches for treasure at a depth of 20 meters below the surface of the sea. At what

  pressure must the scuba device deliver air to the diver? How many atm would this pressure be ? 

              Note :The pressure at the diver’s depth is greater than atmospheric pressure because of the                 weight of the water above the diver. If the air breathed in is not at the same pressure as  

                 the external pressure on the diver’s chest, the excess pressure will collapse the chest. 

                Thus, the breathing apparatus must deliver air to the diver at the pressure of the  

                 surrounding water.  ( use  P = P_atm + h r g )

3.  A tank is filled with water to a depth of  175 cm. What is the pressure at the bottom of the tank  

       due to the water alone ? What is the total pressure ?    

4. Determine the maximum force in newtons exerted by the blood on an aneurysm, or ballooning, by  

 the aorta, given that the maximum blood pressure is 140 mm Hg and the area of the aneurysm is     25 cm².

 F = PA = ( h  r g ) A  = (0.140 m )( 13 600 kg/ m³ ) ( 9.8 m /s²   )( 25 cm² x1 m²/ 10 000 cm²) = 46.648 N

   Note that the fluid involve is mercury, hence we use 13 600 kg/ m³ for the density of mercury  

         and convert 140 mm to meter. The area  25 cm²  is likewise converted to  m² .  

 ****** This is a large force for the vessel to withstand, and there is a considerable risk that the aneurysm  will burst.     

        

5. [ 6.10 / 176 ]  Water towers are used to store water above the level of homes. If a user observes  

    that the static water pressure at home is 3 x 10⁵ N /m², how high above the home is the surface

    of the water in  the tower ?

BUOYANT FORCE AND ARCHIMEDES’  PRINCIPLE

A story has been told that Archimedes ( 287 – 212 B. C. ) conceived of the principle that bears his name after King Hiero of Syracuse asked him to determine the actual composition of the King’s crown, which was alleged to be pure gold. Archimedes was ordered to do so without damaging the crown.

    According to legend, the Greek scientist’s inspiration came to him as he lay partially submerged in his bath. On getting into the tub, he observed that the more his body sank into the tub, the more water ran out over the top. He immediately jumped out of the tub and rushed through the streets naked, shouting excitedly in a loud voice “Eureka” (“I have found it”). Archimedes’ principle states that :

  

          A body whether completely or partially submerged in a fluid, is buoyed upward 

            by a force that is equal to the weight of the displaced fluid.  

                                    Buoyant Force = weight of displaced fluid  ==>  F_B =  w_fl

       The principle applies both to liquids and gases and to objects which are completely or partially submerged.  How this principle allowed Archimedes to solve the problem of the king’s crown is shown in the following problem :

      The King’s crown is said to be solid gold but maybe made of lead and covered  with gold. When it is weighed in air, the scale reads 0.475 kg. When it is submerged in water, the scale reads 0.437 kg.

Is it solid gold ?  If not, what percentage by mass is gold?