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Laboratory experiments Experiment no 7

Investigations of the liquid ow through horizontal pipes

dr Janusz Mi±kiewicz 17 pa¹dziernika 2007

1 Experiment objectives

• Qualitatively investigate Bernoulli's equation.

• Observe the hydrodynamical paradox

• Observe that water do not full ideal liquid requirements

2 Theory

The following knowledge should be acquired:

• Ideal liquid properties

• Continuity equation

• Bernoullie's equation

• Parameters describing uid state

• Pascal law

2.1 Fluid state parameters

Fluid is characterised by the following parameters: Density of the material ρ = ^m_V (mass of the homogeneous material divided by volume)  usually we assume that the liquid is uniform so the density remain constant during the experiment.

Pressure p = ^F_S  force value over the area of the surface on which the force is acting.

1

By the Pascal law we know that: Pressure applied to an enclosed uid is transmitted undiminished to every portion of the uid and the wall of the containing vessel. The hydrostatic pressure of the liquid can be calculated by the equation: p = ρgh, where ρ, g, h are density, acceleration due to gravity, depth respectively.

2.2 Ideal liquid

• The uid is incompressible; its density remain constant.

• The uid do not have frictional eect  it is nonviscous.

• The ow is streamline, not turbulent

• The velocity of the uid does not change during the period of observation (steady- state assumption).

2.3 Continuity equation

The mass of a moving uid does'n change as it ows. If we assume that the density is constant than the later statement can be reformulated as: the volume of a moving uid does not change as it ows. Additionally assuming that the uid speed is constant that we can write the continuity equation in the form:

S1v1= S2v2

The product Sv is volume ow rate. The above equation should be understood: let choose any two surfaces perpendicular to the liquid ow and denote them by number 1 and 2. Than the product of the surface area and velocity of liquid should be the same for all of the chosen surfaces. The important observation is if the tube is narrowing the uid velocity is growing.

2.4 Bernoulli's equation

The Bernoulli's equation can be derived from energy considerations. It say that the pressure plus the total mechanical energy per unit volume p + ρgh + ¹₂ρv², is the same everywhere in a ow tube.

3 The experiment

In this experiment we are using the tube with the part signicantly narrower than the main part of the tube. We measure the dierence between the pressure p1− p2 at the narrow and standard part of the tube. The pressure dierence is measured by mercury manometer. The fraction of pipe's radius is ^r_r¹₂ = 0.1.

During the experiment we measure the volume ow rate and the dierence between pressure measured by manometer.

2

According to continuity equation:

v1S1= v2S2⇒ πr²₁v1= πr²₂v2 (1) Assuming that we write Bernoulli's eq. for the points with the same hight the Bernoulli's eq. takes the following form:

p1+1

2ρv₁²= p2+1

2ρv²₂ (2)

Since we are measuring volume ow rate Q = vS we are going to compare the ow rate measured and calculated from the Bernoulli's eq. From eq.2:

p1− p2= 1

2ρ(v²₂− v²₁) (3)

from continuity eq.1 we can express v1 in terms of v2.

p1− p2= 1

2ρv²₂(1 −r⁴₂

r⁴₁) (4)

From the manometer we get:

p1− p2= (ρ1− ρ)gd (5)

ρ1 - mercury density, ρ - water density, d - dierence in water levels. Calculating velocity of the water from eq.4:

v2= s

2(p1− p2) ρ(1 − (^r_r²

1)⁴ (6)

So the volume ow rate is:

Q = πr₂² s

2(p1− p2) ρ(1 − (^r_r²

1)⁴ (7)

Taking into account that the expression (^r_r²₁)⁴≈ 0we can write:

Q = πr₂² s

2(p1− p2)

ρ (8)

On the other hand the pressure dierence p1−p2is measured by manometer and given by eq.5 so nally

Q = πr²₂ r

2gd(ρ1

ρ − 1) (9)

Since it is dicult to plot square root function we will analyse the square of volume

ow rate:

Q²= 2π²r⁴₂gd(ρ1

ρ − 1) (10)

3

d V t Qo Q v1 v2

m s m^{3 m}_s^{3 m}_s^{3 m}_s ^m_s

let denote k = 2π²r₂⁴g(^ρ_ρ¹ − 1)we get

Q²= kd

We have received the linear dependence of the square of volume ow rate as a function of the dierence mercury levels.

4 Data analysis

Results should be expressed in the form of plot. On the same page we are plotting the measured square of the volume ow rate as a function of mercury level Q²o(d)and square of the volume ow rate calculated from equation 10.

If the experiment was held properly we should be able to t a stride line to the

theoretical and experimental data. The dierence between these two lines should be increasing with the pressure dierence.

The data of the experiment should be collected in the following table:

Additionally the following parameters should be noted:

r₁, r₂, ρ, ρ₁.

4

Laboratory experiments Experiment no 10 Humidity measurement

dr Janusz Mi±kiewicz November 4, 2007

1 Experiment objectives

The main target of the experiment is to measure air humidity. Observe and understand the relationship between air humidity and temperature.

2 Theory

Humidity is feature which describe the amount of water vapour in a sample of air. Humidity can be described by the following parameters: absolute humidity, relative humidity and maximum humidity. The denitions are as follows:

Absolute humidity is the quantity of water in a particular volume of air.

SI units are kilograms per cubic meter [_m^kg3]. In short density of water vapour.

W_b = m_w Va

W_b, mw, Vaare absolute humidity, water mass, air volume respectively.

It does not change with temperature except when the air cools below dewpoint. However, absolute humidity changes as air pressure changes.

Maximum humidity is the maximum amount of water vapour the air can hold at any specic temperature. In short - the maximum density of water vapour which can the water hold at the given temperature.

Relative humidity is denoted as the ratio of the partial pressure of water vapour in a gaseous mixture of air and water vapour to the saturated

1

vapour pressure of water at a given temperature. Relative humidity is expressed as a percentage. The relative humidity can be calculated as a fraction of absolute humidity and the maximum humidity. Both denitions are equivalent.

W_w = p_b pm

100% = W_b Wm

100%

where Ww, Wm, pb, pm are: the relative humidity, the maximum humid- ity, the partial pressure of water vapour, the maximum partial pressure of water vapour.

Dew point (dew-point temperature) The temperature to which a given par- cel of air must be cooled at constant pressure and constant watervapour content in order for saturation to occur. When this temperature is be- low 0^oC, it is called the frost point.

The humidity depends on the air pressure and temperature. However, considering these both parameters from the practical point of view the tem- perature seams to be more important one.

Absolute humidity If the temperature is increasing the absolute humidity is not changing remains constant, but while the temperature is decreas- ing the absolute humidity is not changing until the dew point, further, since the absolute humidity can not be grater than maximal humidity, it is decreasing. if air temperature is decreasing the maximal humidity is decreasing.

Maximal humidity If the air temperature is increasing it is increasing and if air temperature is decreasing the maximal humidity is decreasing.

Relative humidity If the temperature is decreasing the relative humidity is increasing since the maximal humidity is decreasing. The second pos- sibility is exactly opposite it the temperature is increasing the relative humidity is decreasing.

There are various devices used to measure and regulate humidity. A device used to measure humidity is called a psychrometer or hygrometer. A hu- midistat is used to regulate the humidity of a building with a de-humidier.

These can be analogous to a thermometer and thermostat for temperature control.

Humidity is also measured on a global scale using remotely placed satel- lites. These satellites are able to detect the concentration of water in the troposphere at altitudes between 4 and 12 kilometers. Satellites that can

2

measure water vapour have sensors that are sensitive to infrared radiation.

Water vapour specically absorbs and re-radiates radiation in this spectral band. Satellite water vapour imagery plays an important role in monitoring climate conditions (like the formation of thunderstorms) and in the develop- ment of future weather forecasts.

Inuence on human body:

The human body sheds heat by a combination of evaporation of perspira- tion, heat convection to the surrounding air, and thermal radiation. Under conditions of high humidity, the evaporation of sweat from the skin is de- creased and the body's eorts to maintain an acceptable body temperature may be signicantly impaired. Also, if the atmosphere is as warm as or warmer than the skin during times of high humidity, blood brought to the body surface cannot shed heat by conduction to the air, and a condition called hyperpyrexia results. With so much blood going to the external sur- face of the body, relatively less goes to the active muscles, the brain, and other internal organs. Physical strength declines and fatigue occurs sooner than it would otherwise. Alertness and mental capacity also may be aected.

This resulting condition is called heat stroke or hyperthermia.

3 Experiment

The humidity will be measured by two methods:

• Dew point method

• Assman psychrometer

Both methods should not caused signicant diculties. The dew points methods is based on the dew point eect. If we cool the air the maximal humidity is decreasing. We do not observe any eect until the moment when the absolute humidity become equal maximal humidity. Than further decreasing of the temperature result in fog, which is observed on the mirror.

Then we notice the temperature of the mirror and repeat experiment by heating the mirror. The measured temperature allows us to nd the maximal humidity at the temperature of the dew point which is equal to the absolute humidity. In order to calculate the remaining parameters we have to measure the temperature of the air (from the second experimental set) and nd it in humidity tables. In this experiment you are measuring temperature using the thermoelectric eect. In short thermoelectric eect, is the direct conversion of thermal dierentials to electric voltage and vice versa. There is a linear

3

dependency between the temperature dierence and generated potential. For the given experimental set the equation is:

T = U

a + 273[K]

the value of a parameter will be given in the laboratory.

Important!

While measuring the potential you will be using a voltmeter which is not digital and you will be adjusting the range of the meter to the situation observed in experiment. It is useful to notice both sensitivity and position of the pointer on the scale and after the experiment to calculate the dierence of potential.

Assman psychrometer.

In this experiment we are using eect of the dependence between the evapora- tion speed and the air humidity. There are two thermometers one measuring air temperature and one with a wet ending. The temperature of the wet end depends on two eects: one is that evaporation of the water result in decreasing of the material temperature at the same time since the tempera- ture of the wet element is lower than the temperature of the air the energy is transmitted from the air to the wet element of thermometer. If we wait long enough the system will come to the equilibrium where the energy amount lost by evaporation is compensated by the energy transfer. The tempera- ture of equilibrium point depends on the air humidity since the evaporation speed depends on the humidity. The maximal humidity we can nd out from humidity table. The absolute humidity can be calculated by equation:

W_b = W_m⁰ − k(T₁− T₂)

W_m⁰  maximal humidity at the temperature of the wet thermometer, T1, T₂ dry and wet thermometers temperature, parameter k will be given in the laboratory.

4

Figure 1: Absolute humidity tables

4 Data analysis

range U U

mV units mV U_mean T_r T₁ W_r W_w

mV K K _m^kg3 %

Table 1: Measurement table of the dew point method.

T₁ T₂ W_m⁰ W_b W_m W_w K K _m^kg3 kg

m³ kg

m³ %

Table 2: Measurement table of the Assman psychrometer method.

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EXPERIMENT NR. 11

SUBJECT: Surface tension of a liquid

Why do some insects succeed in skating on water instead of sinking? Why in some cases, does the water sprinkled on a glass surface collect into drops and in other cases spread like a thin film? Why does water climb up a thin tube? Why can you make bubbles with soapy water and not with tap water? For reasons we will see later on, the surface of a substance has special properties. These surface properties are what allow these strange phenomena. Not only that, but the surface is also the place of contact among different substances. In short, the properties of surfaces are so special and important that there is a branch of science, the physics of surfaces, devoted to the study of surface phenomena.

SURFACE TENSION

A molecule of a liquid attracts (cohesion) the molecules which surround it and in its turn it is attracted by them (Figure 1). For the molecules which are inside a liquid, the result of all these forces is neutral and all them are in equilibrium by reacting with each other. When these molecules are on the surface, they are attracted by the molecules below and by the lateral ones, but not toward the outside. The resultant is a force directed inside the liquid. In its turn, the cohesion among the molecules supplies a force tangential to the surface. So, a fluid surface behaves like an elastic membrane which wraps and compresses the below liquid. The surface tension expresses the force with which the surface molecules attract each other. A way to see the surface tension in action is to observe the efforts of a bug to climb out of the water. On the contrary, other insects, like the marsh treaders and the water striders, exploit the surface tension to skate on the water without sinking. It is also what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid (as in connecting bits of water or as in a drop of mercury that forms a cohesive ball).

Fig.1. Scheme of the attractive forces among the molecules of a liquid. The inner molecules of surface do not balanced upward and form this originates an inward compression. Besides, the cohesion among the molecules determinates a tension tangential to the surface. So surface of a fluid behaves like an elastic membrane.

Surface tension  has the dimension of force per unit length (eq.1), or of energy per unit area (eq.2). The two are equivalent—but when referring to energy per unit of area, people use the term surface energy—which is a more general term in the sense that it applies also to solids and not just liquids.

(1)

(2)

Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of water tend to be pulled into a spherical shape by the cohesive forces of the surface layer. Another way to view it is that a molecule in contact with a neighbor is in a lower state of energy than if it was not in contact with a neighbor. The interior molecules all have as many neighbors as they can possibly have. But the boundary molecules have fewer neighbors than interior molecules and are therefore in a higher state of energy. For the liquid to

minimize its energy state, it must minimize its number of boundary molecules and must therefore minimize its surface area.

Surface tension strongly depends of the kind of a liquid, it’s contamination and temperature.

Higher the temperature lower the surface tension.

Notions connected with a surface tension phenomenon:

Cohesion (n. lat. cohaerere "stick or stay together") or cohesive attraction or cohesive force is a physical property of a substance, caused by the intermolecular attraction between like- molecules within a body or substance that acts to unite them.

Adhesion is the tendency of certain dissimilar molecules to cling together due to attractive forces. In contrast, cohesion takes place between similar molecules.

Meniscus, plural: menisci/meniscuses, is a curve in the surface of a molecular substance and is produced in response to the surface of the container or another object. It can be either concave or convex. A convex meniscus occurs when the molecules have a stronger attraction to each other than to the container. This may be seen between mercury and glass in barometers. Conversely, a concave meniscus occurs when the molecules of the liquid attract those of the container. This can be seen between water and an unfilled glass. One can over-fill a glass with water, producing a convex meniscus that raises above the top of the glass, due to surface tension.

A meniscus as seen in a burette of colored water. A: The bottom of a concave meniscus B: The top of a convex meniscus

Capillary action or capillarity, refers to certain phenomena associated with the behavior of liquids in thin tubes or in porous materials. Liquids, such as water, will tend to move "up-hill" (against the force of gravity) which does not normally occur in large containers. Capillary action acts on concave menisci to pull the liquid up, increasing the amount of energetically favorable contact area between liquid and container, and on convex menisci to pull the liquid down, reducing the amount of contact area (Fig.2). This phenomenon is important

in transpirational pull in plants. When a tube of a narrow bore, often called a capillary tube, is dipped into a liquid and the liquid ―wets‖ the tube (with zero contact angle ), the liquid surface inside the tube forms a concave meniscus, which is a virtually spherical surface having the same radius, r, as the inside of the tube.

The tube experiences a downward force of magnitude.

When reading a scale on the side of a container filled with liquid, the meniscus must be taken into account in order to obtain a precise measurement. Capillary action occurs when the adhesion to the walls is stronger than the cohesive forces between the liquid molecules. The height to which capillary action will take water in a uniform circular tube is limited by surface tension. Acting around the circumference, the upward force is F_upword =  2r.

The height h to which capillary action will lift water depends upon the weight of water which the surface tension will lift:

(3) The height to which the liquid can be lifted is given by:

(4)

After transformation eq.(4) surface tension is given by:

(5)

Fupword =  2r

Methods for the surface tension estimations

11.1. Detach method

This method is based on the Hooke’s law. Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Hooke's law in simple terms says that strain is directly proportional to stress. Mathematically, Hooke's law states that

(5)

where

x - is the displacement of the end of the spring from its equilibrium position;

FR - is the restoring force exerted by the material; and k is the force constant (or spring constant) in SI units kg/s²

First a very light ring we will place on the surface of the water then we will detach this ring using the restoring force of the spring. The moment when ring will detached water surface refers to equality of two forces, the restoring force FR and the surface force F (Fig.3). Appying eq.1 we will obtain:

FR = F kx =  l

,

where x is an elongation of the spring when we observe detach of the ring. Because we will measure several time this elongation x will change notation to laverage.

So the final equation from which we will calculate surface tension is given by:

FR

F FR

F

Manual of actions:

1. Check the position of the indicator of the spring scales l₀,

2. Put the weight m on the scales’ plate and check the position lk of the indicator, 3. Calculate the spring constant k:

where Δlk =lk – l0

4. Remove the weight from the plate and put the container filled with water under the ring attached to the scales so the ring touches the water (but is not fully immersed).

5. Slowly move the container with water down until the ring detaches from the water. Put down the position l of the scales’ indicator exactly at the moment in which the ring was detaching.

6. The measurement described in the above point should be repeated at least five times.

7. Calculate average l_avg of all measurements of the position l and having it calculate the surface tension coefficient:

where Δlavg =lavg – l0 and r and R is the inner and outer radius of the ring (respectively), values of which are given in the table below

Table 11.1.

r R l0 lk Δlk m k l lavg Δlavg α

m m m m m kg N/m m m m N/m

1.76

·10^-2 1.82·

10^-2

11.2.Capillary Ascension method

Capillary rise method: The end of a capillary is immersed into the solution. The height at which the solution reaches inside the capillary is related to the surface tension by the equation discussed above (eq. 5).

Manual of actions:

1. Because the scale on the capillary pipe is not known it must be measured before starting the experiment. For this use the vernier caliper to measure the distance p between consecutive stops of the capillary pipe’s scale.

2. Put the capillary pipe into the liquid perpendicularly to the surface of the liquid. The pipe should be lifted a bit with its end still remaining in the liquid. The level of the liquid inside the pipe will be higher than the level of the liquid outside.

3. Put down the height of the water inside the pipe expressed in terms of the pipe’s scale (the height of the water is the difference between the level of the water inside the pipe and outside of it).

4. Repeat the measurement at least five times and calculate the average height of the water in pipe.

5. Recalculate the height so it is expressed in meters: h= navg · p, where navg is the average height calculated in the preceding point and p is the distance between pipe’s scale stops calculated in point 1.

6. Calculate the surface tension coefficient using following expression:

where r is the inner capillary pipe’s radius and  is the specific weight of the water – both provided in the table below.

Table 11.2.

p n n_avg h r  α

m m m N/m³ N/m

6.08 · 10^-4 9.81·10³

NAME: DATE:

SURNAME: Hour of the classes

Experiment no 11 – REPORT

Surface tension of a liquid

11.1. Detach method

r R l0 lk Δlk m k l lavg Δlavg α

m m m m m kg N/m m m m N/m

1.76

·10^-2 1.82·

10^-2

Calculate average lavg of all measurements of the position l and having it calculate the surface tension coefficient:

11.2. Ascension method

p n n_avg h r  α

m m m N/m³ N/m

6.08 · 10^-4 9.81·10³

Calculate the surface tension coefficient using following expression:

Laboratory experiments Experiment no 14

Viscosity

dr Janusz Mi±kiewicz November 4, 2007

1 Experiment objectives

The main target of the experiment is to measure viscosity of liquids.

2 Theory

Viscosity is a friction caused by liquids. The viscosity force in a simple case can be dened by Newton's law:

F = ηS∆v

∆y,

where η is the proportionality constant called viscosity, S is the surface area of the plate, ^∆v_∆y - gradient of the velocity calculated in the perpendicular direction to the velocity of the moving surface S.

The viscosity force depends on the object velocity (if the object is moving faster the force is greater), the surface and the viscosity coecient η, which depends on the type of liquid, temperature, density.

2.1 Laminar ow in tube

The rst method uses the dependency between viscosity force and the tube parameters, which is described by the Poisseuille's equation:

V

t = πr⁴∆p

8ηl (1)

1

V

t  ow rate (volume per time),

r  radius of the tube (given in the laboratory),

∆p  hydrostatic pressure, l  the tube length.

In this experiment the Mariot's bottle is used in order to provide the constant liquid pressure during the experiment.

∆p = hρg Substituting it to the eq.1 we get:

η = πr⁴hρ²gt 8lm

2.2 Movement in liquid

This method is based on the movement of a ball in a liquid. If we consider an object in a liquid than there are the owing forces acting on this object:

• gravitation force G = mg = ρoV g

• buoyant force Fb = ρ_bV g

• viscosity force Fη = 6πηrv  in the case of a ball

ρo  density of an object, ρb  dencity of the liquid, V  volume of the ball, v  ball velocity.

The equation of motion from the II dynamic principle:

ma = ρoV g − ρbV g − 6πηrv (2) from the denition of acceleration a = ^dv_dt we have:

mdv

dt = ρ_oV g − ρ_bV g − 6πηrv the solution is:

v(t) = v₀exp(−6πηr

m t) + V g

6πηr(ρ_o− ρ_b)

The velocity of the ball is decreasing until the constant movement.

The same conclusion can be drown from the analysis of the eq.2. If Fη+Fb < G, than the gravitation force is greater and the ball will accelerate, but if its speed will increase than the viscosity force will increase. The process will continue until the system will be in equilibrium and the ball will be in

2

constant movement. On the other hand if Fη + F_b > G the viscosity force will slow down until the ball will reach constant movement.

The measurement is done under assumption of constant movement of the ball. In this case we have Fη + F_b = G, so

η = 2r²(ρ_o− ρ_b)gt

9h .

3

Laboratory experiments Experiment no 18

Measurement of the elastic modulus of bones

dr Janusz Mi±kiewicz October 22, 2007

1 Experiment objectives

The main target of the experiment is to observe the relation between forces and deformation and measure elastic modulus.

2 Theory

The main form of deformation are: Tensile stress  the body is widen by forces acting at its ends as showned at g.1.

Bulk stress  the body is squeezed from all sides by forces, e.g. a body in the water, g.2. Shear stress  while cutting, g.3.

For each kind of deformation can be introduced appropriate quantity called stress that characterises the strength of the forces causing the defor- mation. another quantity, strain, describesthe resulting deformation. When the stress and strain are small enough they are directly proportional. We call the proportionality constant an elastic modulus.

The proportionality of stress and strain is called Hook's law:

stress

strain =elastic modulus.

Figure 1: Tensile stress 1

Figure 2: Bulk stress

Figure 3: Shear stress

The Hook's law in the above form is obeyed in all possible form of stress (the only problem is to dene correctly the stress).

2.1 Tensile and compressive stress and strain

The simplest elastic behaviour is the stretching of a bar, rod, or wire when its ends are pulled (g.1). We restrict the Hook's law analysis to the case of tensile and compressive stress. Figure 4 shows an object that initially has uniform cross-sectional area A and length l0. Then the forces F⊥ are applied of equal magnitudes but opposite directions at the ends. The tensile stress is dened:

tensile stress = F_⊥ A

This is a scalar quantity and is measured in pascals P a = _m^N2. Tensile strain is dened as a relative elongation of the object:

tensile strain = l − l₀ l0

So putting the dened above values into Hook's law we get:

Y = F_⊥l₀ A∆l

Y  Young's modulus Remark: very often Young's modulus is denoted by the letter E.

The case of the compressive stress is described in similar way.

2

Figure 4: Tension and compression

Figure 5: The beam supported at both ends.

Very often, due to the construction reasons, compressive and tensile strain are observed simultaneousely. Eg. beam supported at both ends (g.5) is at the same time under tension (blue colour) and tension (red colour), the middle part (green) is not under tension.

2.2 Elasticity and plasticity

Hook's law  the proportionality of stress and strain in elastic deformation has a limited range of validity. Suppose that we plot a graph as a function of strain. If Hook's law is obeyed , the graph is a straight line with a slope equal to Young's modulus. According to Hook's law deection of the beam h is proportional to the force Fn. A typical example is shown at gure 2.2.

3

The rst portion is a straight line, but it ends at point a; the stress at this point is called the proportional limit.

From a to b, stress and strain are not longer proportional, and Hook's law is not obeyed. If the load is gradually removed, starting at any point between O and b, the curve is retraced until the material returns to its original length. The deformation is reversible  the material shows elastic behaviour. Point b is called the yield point and the stress at this point is elastic limit.

When we increase the stress beyond point b, the strain continues to in- crease. But when the load is removed the material is not returning to its original length. Instead it follows the red line. The material has undergone an irreversible and acquired a permanent set. Further increase of load beyond

c produce a large increase in strain for a relatively small increase in stress, until a point d is reached at which fracture takes place.The behaviour from b to d is called plastic deformation. A plastic deformation is irreversible.

Microscopic point of view. Every material is made of atoms. The solid state exists because these atoms interacts. If you use a force than strain seen from the atomic level is observed as a shift of atoms. However the shift is small and the force is proportional to displacement. When we increase stress beyond elastic limit than the shift of atoms is so big that atoms start to change its location (in respect to the neighbouring particles) in the material.

In results when the stress is removed the particles remain in its new position so the shape of the material has been changed. From this point of view it should be clear why relatively small increase of stress result in signicant increase of strain  because at this limit atoms are very likely to change its position. Of course if we increase the stress too much than the dispal- cement mechanism is not possible to compensate the stress and the material is broken.

3 Experiment

In the experiment you will measure a bone, which can be considered as a beam - g.6.

h = kF_n, (1)

where k is the proportionality coecient, which depend on the size and shape of the section of the beam. In the case of the tube, which is an approximation of the bone empty inside it is:

k = l³

12π(R⁴− r⁴)Y (2)

4

5

Figure 6: Experimental set

Figure 7: Experimental set R and r are radius and inner radius of the bone.

In order to apply a force you will be using a single-arm lever. The system is presented in the g.7. O - pivot point, A - place of bone, B - gravitational force. The force acting on the bone can be calculated as:

Fn = wFg, where w = OB OA In the case of the used experimental set w ≈ 3.

4 Data analysis

1. Measure distances OA, OB and calculate w = ^OB_OA. 2. Measure bone length. Measurement repeat 5 times.

3. Place the dial micrometer. Measure deection of the bone using 7 dierent loads.

4. Measure 10 times bone diameter (D) and 3 times inner diameter (e).

5. We make the plot of the deection as a function of the load.

6

6. From the plot calculate linear coecient k - equation 1 and use it to calculate elastic modulus from equation 2.

Data tables

l lm w Fg F_n h D Dm e em d_m r R 1/Y

m m N N m m m m m m m m ^m_N²

h Fn

m N

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Laboratory experiments Experiment no 21

Blood transport

dr Janusz Mi±kiewicz December 2, 2007

1 Experiment objectives

In this experiment you have to apply the thermodynamic principles in order to measure ow rate on the body. Particularly through the hand. The exper- iment is based on the I thermodynamic principle and using heat equation.

From the second thermodynamic principle we know that the energy is transported from a body with higher temperature to the body with lower temperature. We can calculate the transfered energy knowing initial and nal temperature as well as mass of the body. Of course the opposite conclusion is also possible: knowing the energy transfered and the dierence of the nal and initial temperature we can calculate the mas of an object - here blood.

1.1 Experiment idea

In the experiment take into consideration the following elements: calorimeter, water, hand body, blood transfered through an arm.

Elements accepting energy are: calorimeter, water.

Elements providing energy is blood.

The body of the hand plays a special role: initially it is providing energy, but after approximately 10 minutes it is accepting energy.

The parameter measured are: volume of the hand, mass of the water, tem- perature of water (which is assumed to be identical to the whole system).

The main assumption are:

• blood, which is entering hand has the human body temperature (37^oC)

• leaving hand has temperature of the water bath.

1

These two assumptions allows us to compare the energy provided by blood to the system with the mass of blood.

2 Theory

The rst law of thermodynamics: The energy of a system can be changed by heat transfer ∆Q or by some work ∆W :

∆U = ∆Q + ∆W

When there the work ∆W = 0 then we can simplify the rst law of thermodynamics and write the heat equation. Writing the heat equation we have to distinguish subsystems which exchange thermal energy. In the case of two subsystem the heat equation takes the form:

∆Q1 = ∆Q2

m₁c₁(T₁− T_f) = m₂c₂(T_f − T₂) (1) where: m1, m2, c1, c2, T1, T2  mass, specic heat, initial temperature of the body 1 and 2, Tf nal temperature of the system.

Eq.(1) is valid when the temperatutures are well dened. Unfortunately in this experiment the water temperature is constantly changing as well as the nal temperature of the blood. So while we can measure the change the water and calorimeter internal energy and to know how much heat was provided be blood to the water it require some caution while calculating the mass of the blood. In order to nd the mass of the blood we have to analyse the process of heat transfer in details - in short time intervals. If we choose very short time intervals we can neglect the change of the nal temperature of blood and write:

∆Q(t + ∆t) = V_od_krc_kr(37^o− T (∆t))∆t

d_kr  blood's density, ckr  blood's specic heat, Vo  blood ow rate.

Since we know the change of the water energy we have to calculate the total energy provided by blood by summing up ∆Q(t + ∆t) from the time 10 min up to 30 min. The rst moment (10 min) is the time required to cool down the hand to the temperature of the surrounding water. Of course in order to do the calculation properly we have take the limit ∆t → 0. Finally we get:

∆Q =

Z _30min

10min V_od_krc_kr(37^o− T (t))dt 2

rewriting

∆Q = V_od_krc_kr(20min)(37^o− 1 20min

Z _30min

10min −T (t)dt (2)

introducing the mean temperature Tm = _20min^{1 R}_10min^30min−T (t)dt we can write the eq.(2) in simple form:

∆Q = V_od_krc_kr(20min)(37^o− T_m)

The only problem in the above equation is to nd the mean temperature.

Instead of calculating the integral we can measure the surface below the curve (which is equivalent the calculating the integral) and divide it by the time interval (here 20min).

On the other hand the change of the energy of the water, calorimeter and hand can be calculated as:

Q_p = (V_rdc_r+ m_wc_w+ P_k)(T₃₀− T₁₀) (3) V_r hand's volume, cr specic heat of human body, mw  mas of the water, cw  specic heat of the water, d  human body dencity, Pk  heat capacity of the calorimetry. Comparing eq.(2) and eq.(3) we can nd out the blood rate:

V_kr = (V_rdc_r+ m_wc_w+ P_k)(T₃₀− T₁₀) d_krc_kr(20min)(37^o− T_m) and

V_o = V_kr 20min

Knowing the ow rate Vo, volume of a hart W = 70cm³ and the pulse n we can calculate the fraction of the blood going through the hand:

p = V_o W n100%

3

EXPERIMENT NR. 25

SUBJECT: Resistance measurements

The electrical resistance of an object is a measure of its opposition to the passage of a steady electric current. An object of uniform cross section will have a resistance proportional to its length and inversely proportional to its cross-sectional area, and proportional to the resistivity of the material.

Discovered by Georg Ohm in 1827 electrical resistance shares some conceptual parallels with the mechanical notion of friction. The SI unit of electrical resistance is the ohm (Ω).

For a wide variety of materials and conditions, the electrical resistance does not depend on the amount of current through or the potential difference (voltage) across the object, meaning that the resistance R is constant for the given temperature and material. Therefore, the resistance of an object can be defined as the ratio of voltage to current, in accordance with Ohm's law:

A solid conductive metal contains mobile, or free, electrons. These electrons are bound to the metal lattice but not to any individual atom. Even with no external electric field applied, these electrons move about randomly due to thermal energy but, on average, there is zero net current within the metal. Given a plane through which the wire passes, the number of electrons moving from one side to the other in any period of time is on average equal to the number passing in the opposite direction. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons.

When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current. For a steady flow, the current I in amperes can be calculated with the following equation:

where Q is the electric charge in coulombs [C] transferred, and t is the time in seconds (s).

More generally, electric current can be represented as the time rate of change of charge, or

Direct Current resistance

The resistance R of a conductor of uniform cross section can be computed as

where, l is the length of the conductor, measured in metres [m], A is the cross-sectional area of the current flow, measured in square metres [m²] ρ is the electrical resistivity (also called specific electrical resistance) of the material. The SI unit of electrical resistivity is the ohm

metre [Ω m]. Resistivity is a measure of the material's ability to oppose electric current. For practical reasons, any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductors such as wires.

Electrical resistivity (also known as specific electrical resistance or volume resistivity) is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electrical charge.

Electrical resistivity ρ is defined by,

where, ρ is the static resistivity (measured in volt-metres per ampere, V m/A); E is the magnitude of the electric field (measured in volts per metre, V/m); J is the magnitude of the current density (measured in amperes per square metre, A/m²). The electrical resistivity ρ can also be given by,

where R is the electrical resistance of a uniform specimen of the material (measured in ohms, Ω); l is the length of the piece of material (measured in metres, m); A is the cross-sectional area of the specimen (measured in square metres, m²).

Finally, electrical resistivity is also defined as the inverse of the conductivity σ (sigma), of the material, or

The reason resistivity has the units of ohm-metres rather than the more intuitive ohm per metre (Ω/m) can perhaps best be seen by transposing the definition to make resistance the subject;

The resistance of a given sample will increase with the length, but decrease with the cross sectional area. Resistance is measured in ohms. Length over Area has units of 1/distance. To end up with ohms, resistivity must be in the units of "ohms × distance" (SI ohm-metre)

The aim of this experiment is to estimate electrical resistance by using two method:

1) less precise called “technical method”

2) more precise Wheatstone bridge.

Technical Method

This method is directly based on the eq.1. To perform this part of the experiment you would need: voltage source, amperometer and voltameter, set of connecting wires and wirewound resistors with an unknown electrical resistance R_x which must be measured. All these elements we may connect into electrical circuit which is a network that has a closed loop, giving a return path for the current. A number of electrical laws apply to all electrical networks. These include: Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff.

Kirchhoff's current law:

The sum of all currents entering a node is equal to the sum of all currents leaving the node. At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node. Adopting the convention that every current flowing towards the node is positive and that every current flowing away is negative (or the other way around), this principle can be stated as:

n is the total number of branches with currents flowing towards or away from the node.

Kirchhoff's voltage law:

This law is also called Kirchhoff's second law. The directed sum of the electrical potential differences (voltage) around any closed circuit must be zero. Similarly to Kirchhoff's current law, it can be stated as:

Back to the experimental part. Already mentioned elements voltameter, amperometer, power supply and resistor Rx will be connected into the (A) and (B) type of the electrical circuits.

In the “technical method A” voltameter shows voltage on the measured R_x, so V = V_x, where V – is a voltage read from the voltameter and Vx - is voltage on the resistor Rx.

Amperometer shows current (J) which is the sum of the currents flow both through resistor Rx

(J_x) and voltameter (J_v), so according Kirchhoff's current law:

If Rx<< Rv, then according Kirchhoff's voltage law Jx>>Jv, so we can assume that:

Finally we will obtain:

In the “technical method B” voltameter indicates not only voltage on the resistor Rx, (Vx) but also drop of the voltage on internal resistor of the amperometer, V_A;

So, according eq.1 and assumption that J = Jx we will obtain:

where is an internal resistance of the amperometer.

When R_x>>R_A we can neglect R_A and calculate R_x from the eq.5.

Tasks for the technical method A and B:

1. Please connect circuit A and B, respectively.

2. Please ask supervising person to check electrical circuit.

3. Then please set voltage values and read values of the currents. Please measure at least twice for two different values of the voltage.

4. Please calculate Rx from the eq.5.

Important!!! Please take into account if R_x >> R_A or R_x << R_A

Table for the technical method A and B:

Method

number of the resistor

J V RA Rv Rx

[A] [V] [ ] [ ] [ ]

A

B

Wheatstone bridge method

For this method we have to have: voltage source, set of connecting wires, wirewound resistors with an unknown electrical resistance Rx which then must be measured, galvanometer (G), resistor with adjustable values (Rz), slat with a stretched wire which then will be divided by the slide D into resistors R₁ and R₂. Please see scheme below.

The Wheatstone bridge is a measuring instrument invented by Samuel Hunter Christie in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. It is used to measure an unknown electrical resistance Rx by balancing two legs of a bridge circuit, one leg of which includes the unknown component.

Rx is the unknown resistance to be measured; R1, R2 and the resistance of Rz are adjustable. If the ratio of the two resistances in the known leg (R₂ / R₁) is equal to the ratio of the two in the unknown leg (Rx / Rz), then the voltage between the two midpoints (B and D) will be zero and no current will flow through the galvanometer G. Therefore, if R1, R2 and Rz are known to high precision, then R_x can be measured to high precision. Very small changes in R_x disrupt the balance and are readily detected.

When the voltage between two midpoints B and D is equal zero first, Kirchhoff's first rule is used to find the currents in junctions B and D:

J_x – J1 + J_G = 0 J_Z – J₂ – J_G = 0

Then, Kirchhoff's second rule is used for finding the voltage in the loops ABD and BCD:

JxRx – J1R1 + JGRG = 0 J_ZR_Z – J₂R₂ – JGR_G = 0

The bridge is balanced and

J

= 0

JxRx = J1R1

J_ZR_Z = J₂R₂

Then, the equations are divided and rearranged, giving:

From the first rule,

I

= I

I

= I

R

At the point of balance, the ratio of

R

/ R

= R

/ R

Here, resistances R1 and R2 are resistances of pieces of the wire which is stretched on the wooden bar. Using eq. 3 we obtain:

Tasks for the Wheatstone bridge method 1. Please connect circuit.

2. Please ask supervising person to check electrical circuit.

3. From the method A and B please calculate values of the R_x. These values will be then used to preset the Rz resistance for the Wheatstone bridge method.

4. Then please move slide D to balance the Wheatstone bridge. Please note that galvanometer has adjustable protective resistor R. To read correct values of l1 and l2 the potentiometer of the resistor R has to be set at the minimum!

5. Please calculate R_x values using eq. 6.

Number of resistor

l1 l2 RZ RX

[m] [m] [ ] [ ]

Comparison Table:

6. Please calculate percentage deviations according following formula:

Where Rx (WB) is the value of Rx resistance obtain by the Wheatstone bridge method and Rx (A or B method) is the value of R_x resistance obtain either by A or B technical methods.

Scheme in the technical

method

Number of resistor

PD

[ ] [ ]

A

B

A

B

7. Please write a conclusions according result obtained from the comparison both technical and Wheatstone bridge method.

NAME: DATE:

SURNAME: NUMBER of the team:

Experiment no 25

Subject:

Resistance measurements

Table of measurements and results – the technical method A and B:

Method

number of the resistor

J V R_A R_v R_x

[A] [V] [ ] [ ] [ ]

A

B

NAME: DATE:

SURNAME: NUMBER of the team:

Experiment no 25

Subject:

Resistance measurements

Table of measurements and results – Wheatstone bridge method Number of

resistor

l1 l2 RZ RX

[m] [m] [ ] [ ]

Comparison Table:

Scheme in the technical

method

Number of resistor

PD

[ ] [ ]

A

B

A

B