Physics For Premedical Students

Sheet # 3 Properties of Matter

Dr. Maan A. Ibrahim

2008 - 2009

Fluid Pressure

Pressure

The pressure is : the force that is applied over a given area.

SI = Pascal (Pa)

Basic unit = N/m²

= kg/m.s²

With same force exerted on two different areas bigger pressure with smaller

area

(SI unit: pascal (Pa)=N/m2; 1KPa=1000Pa)

Block cannot pop the balloon but the pin can pop up

the balloon. b/c pin has a very small area big

pressure

ex) A weight of 4000N box of cube shape. Each side of the square bottom is

2m.. What pressure does the box exert on the ground?

4000N ÷(2m×2m)=4000N÷4m2=1000N/m2=1000Pa=1KPa

Fluid pressure

Static pressure in fluids: if fluid is at rest (not flowing or moving) pressure is

equal in all directions Pascal’s Principle

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water pressure increases when depth increases.

Pascal's Principle - pressure in fluid (liquid/gas) gets evenly distributed

hydraulic press (ex car lift in garage)

P = F1/A1 = F2/A2 so, by increasing A2, you can increase F2

Pressure of a static (=not moving) fluid (hydrostatic pressure)

Liquid Pressure   = density × gravitational acceleration × height(=depth)

P is the hydrostatic pressure (in pascals);

ρ is the water density (in kilograms per cubic meter); 1g/cm3 = 10-3kg/10-

6m3=1000kg/m3

g is gravitational acceleration (in meters per second squared);

h is the height of fluid above (in meters).

Pressure on Dams

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Liquid Pressure   = density x gravitational acceleration depth

P1 = 1000 kg/m3 9.8m/s2 x 3m P2= 1000 kg/m3 9.8N/kg x 6m

= 1000 kg/m3 9.8N/kg x 3m = 58,800N/m2= 58,800Pa = 58.8Kpa

= 29400N/m2=29000.4Pa =29.4 KPa

When a body is submerged in a fluid, the fluid exerts a force perpendicular

to the surface of the body at each point on the surface. However, for a fluid, the

force does not necessarily need to be a constant over the whole surface. This leads

us to a more refined definition of pressure. We define the pressure p at a point in a

fluid as the ratio of the normal force dF on a small area dA around that point, to the

area

From this definition we can immediately see that pressure also has units of Pascals.

The pressure due to a fluid tends to compress a body. The ratio of the pressure to

the fractional decrease in volume is called the bulk modulus

The inverse of the bulk modulus is called the compressibility, k.

Air Pressure and the Atmosphere

air pressure depends on the depth/height of air on top of us. (=altitude)

altitude increases (when you go high) pressure decreases (b/c height of air

decreases)

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Pascal vase – several shape vases all connected at the bottom

When water put in, water flows until all vases have same height (=depth) of water in all vases

Pressure depends on height, not on amount of fluid

Normal atmospheric pressure (normal air pressure at sea level)

1 atmosphere = 760 mmHg= 29.92 in Hg = 14.7 lb/in2 = 101,300Pa =101.3 KPa

= 1,013.0 millibar

(barometer)

Fluid Mechanics

Let us now look at the properties inherent in a fluid. A fluid differs from a

solid in that it cannot support a shear stress. In this sense, both liquids and gases

can be described as fluids (in fact, under certain extreme conditions, even solids

exhibit fluid-like properties).

The density of a fluid is defined as its mass per unit volume

Density is considered to be inherent to the fluid and can be used to characterize it.

The specific gravity of a fluid is the ratio of its density to that of water. It is a

dimensionless number. Since the density of water is defined to be 1 gm/cm 3, the

specific gravity of a fluid is the density without the units, if the density was also

given in gm/cm3.

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Properties of Matters

1) Density

Mass density is the mass per unit volume.

The volumes of different matters are equal, but they aren’t equal in their masses.

Density ρ=

The density’s unit in SI unit and in the basic unit are same = kg/m³

The density depend on:

1- The kind of matter.

2- Temperature.

3- Pressure.

The density of solids is higher than the density of liquids and the density of liquids

is higher than the density of gases.

The densities of solids and liquids are approximately independent of pressure.

Effect of temperature on the Density:

Most substances expand when their temperature increase , but the densities of

substances decrease , the relation is inversely proportion Accept the water , is

exempted between temperatures 0 and 4 °C.

Specific Gravity of substance:

SG (Specific Gravity) =

kg/m³

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Atmospheric Pressure:

The weight of the air in the upper of the Earth’s atmosphere exerts pressure on

the layer of the air below .

Mercury barometer

1 atm = 1.013 x =760 mmHg

Pressure in a fluid

Pressure is defined as the perpendicular force on a surface per unit surface

area.

[N/m2 = Pascal = Pa]

Mass density is mass per unit volume.

[kg/m3, g/cm3, …]

Water has a density of about 1 g/cm3 (= 103 kg/m3).

Due to gravity, pressure increases with depth in a fluid as

,

where P0 is the pressure at the top of the fluid and P is the pressure at a depth h.

As seen in the diagram to the right, additional pressure below a section of a fluid is

required to keep this section from sinking. Since this section of fluid is in

equilibrium,

But M = V = Ah, so

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If the fluid container is open to the atmosphere, then P0 is atmospheric pressure.

At sea level

P0 = 1.013 x 105 Pa ( = 14.7 lb/in2 = 1 atm)

Example:

What is the downward force exerted by the atmosphere on top of a 2 m x 1 m desk

top.

F = P0A = (1.013 x 105 Pa)(2 m2) = 1.026 x 10 5 N

The reason why this enormous force doesn’t crush the desk is because of a nearly

equal upward force on the bottom of the table.

Example:

At what depth in water is the pressure 2 atm?

Example:

A mercury manometer consists of an inverted tube of mercury as shown to the

right. The top end is closed and the void at the top is essentially a vacuum. The

bottom end is open and is in an open container of mercury. What is the height of

the column of mercury in the tube? The specific gravity of mercury is 13.6.

Hydraulic press

A hydraulic press uses a fluid to magnify an applied

force. A force F1 applied to the small piston of area A1

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increases the pressure in the fluid by P1 = F1/A1. This pressure increase is

transmitted uniformly throughout the fluid (Pascal’s principle). This additional

pressure results in a lift on the large piston.

Example:

In a hydraulic press, the diameter of the small piston is 2.5 cm and the diameter of

the large piston is 10 cm. If the force applied to the small piston is 500 N, what is

the force applied to the large piston?

Archimede’s Principle

Archimede’s principle states that an object submerged in a

fluid is buoyed up by a force equal to the weight of the fluid

displaced by the object.

The buoyant force, B, is just a consequence of the fact that the pressure below the

object is greater than above it. To understand Archimede’s principle, we envision

replacing the object with fluid of the same size and shape. This fluid must be in

equilibrium and have the same buoyant force as the object. Thus, its weight and

the buoyant force must be the same.

Example:

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A cubical block of aluminum 10 cm on edge is suspended in water by a cord.

What is the tension in the cord? The density of Al is 2.7 x 103 kg/m3.

Since the block is in equilibrium,

Example:

An ice cube floats in a glass of water. What fraction of its

volume is below water? The specific gravity of ice is

0.917.

Thus, 91.7% of the volume of the ice is below the water.

Fluid Dynamics

Equation of continuity

The rate at which fluid mass flows through two different parts of the same pipe

must be the same. Thus,

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If the fluid is incompressible, i.e., its density is nearly the same throughout the

pipe, then we have

(Eq. of continuity)

Example:

A hose of diameter 3 cm has a nozzle of diameter 1 cm. If the water flows at 2 m/s

in the hose, what is the water speed as it goes through the nozzle?

Bernoulli’s equation:

Bernoulli’s equation gives a relationship in a flowing fluid between the fluid’s

pressure, flow speed, and elevation. It is based on conservation of energy and

holds for an ‘ideal’ fluid. The ideal fluid would be (1) non-viscous, (2)

incompressible, (3) steady in its flow, and (4) non-turbulent.

Bernoulli’s equation

This means that if you pick any two points in a flowing fluid,

where p is the absolute pressure. This is called Bernoulli's Equation.

If the fluid is at rest (v1 = v2 = 0), then Bernoulli’s equation is the same as the

earlier equation giving P as a function of depth in a fluid –

Qualitatively, Bernoulli’s equation says that the pressure is lower in a region of a

fluid where its speed is greater.

Example:

An airplane wing has curvature and angle of attach such that the air speed above

the wing is greater than below. If v(below) = 100 m/s and v(above) = 103 m/s and

the area of the wing is 10 m2, what is the lift on the wing? The density of air is

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about 1.3 kg/m3. The difference in elevation below and above the wing is nearly

the same, so y1 ~ y2 and

Example:

Water enters a house through a pipe 5 cm in inside diameter at absolute

pressure of 1 x 105 Pa. The pipe leading inside the house to the second floor 2.5 m

above has an inside diameter of 3 cm. When the flow velocity at the inlet pipe is

1.5 m/s, what is the flow velocity and pressure at the upstairs bathroom?

Solution :

From Bernoulli's equation, the pressure is

Properties of a System

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Density, , is defined as the mass per unit volume

alternatively, we may define the specific volume as the volume per unit

mass:

Pressure: An intensive property of a fluid, defined as the average force

exerted by the fluid per unit area:

P = Force/Area

Atmospheric Pressure: The atmosphere surrounds Earth for a depth of

about 70 kilometers. The weight of this air exerts a force at sea level of

about:

Patm = 1 atm. = 14.7 lbf/in2 (psi) = 101 kN/m2 (kPa) = 1.01 bars

The atmospheric pressure is reduced as we climb to higher elevations, in

relation to the decrease in the weight of air above the surface.

Absolute Pressure: If we increase the force exerted on a fluid, the resultant

pressure is increased. The pressure on the gas, measured in this fashion, is

said to be the absolute pressure. Most scientific measurements are based on

absolute pressure measurements.

Gauge Pressure: In many common applications, it may not be convenient to

determine the atmospheric pressure. In such applications gauge pressure is

often used:

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W Pabs = (W+PatmA)/A

A

Patm

Pabs = (W+PatmA)/A

Pgauge = W/A

Pgauge = Pabs - Patm

Pvaccuum = Patm - Pabs

Manometers

Manometers are simple, inexpensive and accurate devices used to measure

fluid pressure.

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W Pabs = (W+PatmA)/A

A

Patm

Pabs = (W+PatmA)/A

Gasz

PA = 1 atmA

B PB = PA + m·g/A = PA + [(A·h)] g/A

= PA + ·g·z

Because of the low density of the gas, the gas pressure will be virtually

uniform within the container. We see then that the differential height

of the fluid in the column is a direct measurement of the gauge

pressure of the gas :

Pgauge = ·g·z

Example :

Assume that the manometer fluid is an oil with a specific gravity of

0.87 and that we can read elevational differences of 0.001 m (1 mm).

Solution :

Pgauge = (0.87 x 1000 kg/m3)(9.81 m/s2)(0.001m)

= 8.53 Pa

= 8.42x10-5 atm

= 1.24 x10-3 psi

fluid

FLUID PROPERTIES

Basic Units

Every fluid has certain characteristics by which its physical condition may be

described. These characteristics are called the properties of the fluid.

Properties are expressed in terms of a number of basic dimensions (length, mass

or force, time, temperature) which are quantified by basic units.

There are two systems of units: the SI system of units and the traditional units.

SI System of Units

The basic units of mass, length, time, and temperature are the kilogram

(kg), meter (m), second (s) and Kelvin (K).

The dimensions and units of other quantities are derived from those of the

basic dimensions and units, e.g.

Specific Gravity, S

The ratio of the specific weight of a given fluid to the specific weight of

water at 4oC is defined as the Specific Gravity.

The specific gravity is dimensionless. Why ?

Ideal Gas Law

The equation of state for an ideal gas is :

where P = absolute pressure; T = absolute temperature = density; R = gas

constant.

The gas constant is related to the universal gas constant Ru by

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where M = molar mass. (Note: Ru = 8.314 KJ/Kgmol . K)

Viscosity

The term viscosity (also referred to as the body of paint) describe the liquid

paint’s resistance to flow.

Viscosity results from the attractive forces that occur between the particles

that make up paint. Attractive forces not only prevent particle from separating

from one another but also restrict the movement of the body of liquid itself.

Because attractive forces decrease with increasing temperature, temperature has a

strong effect on paint viscosity.

viscosity - the resistance of a liquid to flow; the higher the viscosity, the more

sluggish to flow

Viscosity arises from the forces between molecules; strong intermolecular forces

hold molecules together and do not let them move past one another.

Viscosity usually decreases as the temperature rises. Molecules have more energy

at high temperatures and can wriggle past their neighbors more readily.

The viscosity of water at 100 oC is only one-sixth of its value at 0 oC.

Viscosity usually decreases with increasing temperature.

In a viscous fluid, the shear stress is proportional to the time rate of strain.

Mathematically it is expressed as

where is the shear stress, V is the fluid velocity and y is the distance

measured from the wall. dV/dy is the rate of strain (which is also the

velocity gradient normal to the wall). The proportionality factor, , is the

dynamic, or absolute, viscosity. It is a property of the fluid.

The SI unit of viscosity is N.s/m2.

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Consider the velocity distribution next to a boundary. The following

observations can be made:

The velocity gradient becomes smaller with distance from the boundary.

Therefore, the maximum shear stress is at the boundary.

The fluid velocity is zero at the stationary boundary. Viscosity causes the

fluid to adhere to the surface. This is known as the no-slip condition.

Kinematic viscosity is defined as

The viscosity of a gas increases with the temperature of the gas. The variation of

gas viscosity with absolute temperature can be estimated with Sutherland’s

equation,

where o is the viscosity at temperature To and S is Sutherland’s constant.

The viscosity of a liquid decreases as temperature increases. An equation for the

variation of liquid viscosity with absolute temperature is :

where C and b are constants.

Surface Tension

The molecules at the surface of a liquid have a greater attraction for each other

than they do for molecules below the surface. The surface behaves as if it were a

“skin” or “membrane” stretched over the fluid mass.

Because of the membrane effect, each portion of the surface exerts “tension” on

adjacent portions of the surface or on objects that are in contact with the liquid

surface.

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The tension acts in the plane of the surface. The magnitude of the tension per

unit length is defined as the surface tension, The concept of surface tension has

been used to explain several commonly observed phenomena. Examples of these

are:

A steel needle will float on water if placed gently on the surface because

the surface tension force supports the needle.

Capillary action in a small-diameter tube .

The excess pressure created inside droplets and bubbles.

Vapor Pressure

The pressure at which a liquid will boil is called its vapor pressure.

The vapor pressure increases as the temperature increases.

Illustration : At 100oC, the vapor pressure of water is 101.3 kPa, whereas at 10oC

the vapor pressure is 1.23 kPa. This means that water at 101.3 kPa will boil at

100oC. But if the pressure of water is lowered to 1.23 kPa, boiling will take

place at 10oC.

Application : In flowing fluids it is possible to have very low pressure due to the

fluid motion. The liquid will boil if the pressure is lowered to the vapor pressure.

This may occur in flow through the irregular, narrowed passages of a valve or

the suction side of a pump. The vapor bubbles formed will collapse in regions of

higher pressure downstream. This phenomenon (i.e. the formation and

subsequent collapse of vapor bubbles) is called Cavitation. It can lead to

structural damage and, therefore, should be avoided.

The weight of the atmosphere exerts a pressure of 14.7 pounds per square

inch of force at sea level.

This is to say that a 1 inch column of air as tall as the atmosphere, would

weigh 14.7 pounds. We commonly call this .

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1 Atmosphere of pressure, or 1 Atom.

(=101.3 kPa the SI unit)

Water exerts much more pressure. It only takes a 1 inch column of sea water

33 feet tall to weigh 14.7 pounds.

This means that at a depth of 33 feet deep in the ocean, there is a total

pressure of 29.4 pounds per square inch (psi). This would be 2 ATMs of

pressure. One ATM from the water, + one ATM from the air. We call this the

ambient pressure or absolute pressure.

Absolute pressure differs from gauge pressure. Gauge pressure would be the

pressure that would show up on a gauge at this depth. A pressure gauge would start

at 0 at the surface and show 14.7 psi at our depth of 33 feet. Gauge pressure would

ignore the 14.7 psi of atmospheric pressure.

Continuity Equation

Consider the pipe section shown above. The pipe cross sectional area changes from

A1 to A2. The question here is what happens to the fluid velocity.

If the pipe is not leaky and the fluid does not compress, then :

we can calculate each of these sides. Also since the fluid does not compress, the

density is constant throughout the fluid. Thus: in a time t, we have:

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This is easily solved to give:

This is known as the “Continuity Equation” for fluids.

Next, you may have wondered just how a fluid flow works (yes... they are pretty

clear things right up to the point that you start to siphon a fluid).

undiminished through the fluid.

Let’s see how this all works: If you place a weight say m1 on plate A1, then the

pressure pulse is given by:

According to Pascal’s principle, this overpressure must be the same at plate 2.

Thus, at plate 2,

This overpressure will lift a weight m2. How much will it lift?

The essence is that the smaller A1 is, the bigger the mass m2 that can be lifted. This

is basically how a hydraulic lift or a hydraulic press works.

Solids and Fluids

This chapter deals with properties of solids and fluids (liquid and gases). One of

the properties of a material is its elasticity, which is a measure of the extent to

which a force can deform the material. In particular, the elastic modulus, is

defined as

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The elastic modulus is a measure of the stiffness of a material. Stress is the

applied force per unit area (N/m2 = Pascal) and strain is the fractional change in a

dimension due to the stress. There are three main types of elastic moduli.

Young’s modulus – elasticity of length

Up to a certain limit, the stress is typically proportional to the strain; i.e., Y is a

constant. Beyond a certain stress called the elastic limit, this proportionality ceases

to exist and the material will be irreversibly strained. Additional stress will

eventually break the material.

Shear modulus – elasticity of shape

Bulk modulus – elasticity of volume

P is the change in the pressure (force per unit area normal to the surface). The

minus sign in the above equation is because an increase in pressure gives a

decrease in volume and B is positive.

Example:

A copper wire has a length of 1 m and a diameter of 2 mm. How much force is

required to stretch the wire by 1 mm? Y(copper) = 11 x 1010 Pa.

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Problems and Answer

1- What is the mass of the air in a living room has dimension of 4.0m wide, 6.0m

length, and 2.5m height ?

Solution:

(Mass) m= ?

ρ= → m= ρ.V

(volume ) V = wide x length x height

= 4.0m x 6.0m x 2.5m = 60 m³

ρ of the air = 1.29 kg/m³

m= ρ.V = 1.29 kg/m³ x 60 m³ = 77.4 kg

2- If you have a cube of gold its length 6 cm and its mass 4168 g. Find the

density of gold?

ρ =

m= 4168 g = 4168 x kg = 4.168 kg

V= = 216 cm³ = m³ = 2.16 x m³

ρ = = = 19296.3 kg/m³

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3- What volume of water has the same mass as 2.0 m³ of lead ?

Solution:

V of lead = 2 m³

ρ of lead = 11.3 x kg/m³

m of lead = ρ.V = (11.3 x = 11.3

mass of water = mass of lead

m of water = 11.3

ρ of water= 1.00 kg/m³

V of water = 11.3 m³

4 – the nucleus of uranium has a diameter of 1.5 x and a mass of 4 x

kg. What is the density of the nucleus ?

Solution:

ρ =

m= 4 x kg

Sphere volume = V= r³

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Radius (r) = = 0.75 m

(Prefixes form)

= 0.75 x

= 0.75 (10 x ) = 15-

= 7.5 = 7.5 fm

V = (0.75 ³ = 1.767

ρ =

5-The dimension of rectangular of wood is 10m x 8m x 6m. what is the mass of

the rectangular if its density is 0.6 g/cm³ ?

Solution:

M= ? V= 10 8 6= 480 m³

ρ = = = 6 x

m= ρ.V = 480 (6 x = 288

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6- How much is the pressure at a depth of 11.0 km in the Pacific Ocean . Assume

the density of sea water is 1.025 g/cm³ and the atmospheric pressure is 1.01 x

Pa ?

Solution:

h = 11.0 km = 11 x m

ρ = 1.025 g/cm³ = 1.025 x kg/m³

P = ρ h g = (11 x (1.025 x ) x 9.81

= 1.106 x Pa

7- Two fish swim in the fresh water lake, oneا at a depth of 14 m and the other at a

depth of 98.0 m. What is the different in pressure on the fish? Select the

reasonable values for P and ρ?

Solution:

ρ = 1.025 x kg/m³

freshfirst

P1 = ρ h1 g

= 1.025 x x 9,81

= 1.41 x Pa

Second fresh

P2 = ρ h2 g

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h1=14mh2=98m

= 1.025 x

= 9.85 x

P = 1.01

P total = P + P2 = ( 1.01 x ) + (1.41 x

= 1.42 x Pa

Pt = P + P2 = ( 1.01 x ) + (9.85 x

= 9.956 x Pa

8- Suppose a plunger has a cross sectional area of 0.1 m2 while a second plunger

has an area of 0.01m2. The system is connected together via yucky orange fluid.

How much weight will the large plunger be able to pick up if 10 kg is placed on

the small plunger?

Solution:

let 1 be the smaller plunger and let 2 be the larger plunger. Then, according to

Pascal’s principle:

The ratio of areas is a factor of 10. Thus,

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9- Suppose you eat lots of bad food that eventually blocks the cross sectional area

of an artery by a factor of ½ . How much lower is the blood pressure in this region

of the artery than an unblocked region ?

Solution:

The continuity equation tells us how much faster the fluid flows in the blocked

region. Let 1 be unblocked and 2 be blocked. Then :

10- Write the values in the prefixes form ?

Pressure in

Pa

Pressure in prefixes form

2 x 2 x = 20 P Pa

4 x 4 x 400 x 400 G Pa

6 x 6 x = 60 x

1.01 x 1.01 x K Pa

2.8 x 2.8 x

1 x = 1 p Pa

Problems

1- The prefixes used to indicate multiplication of units are :

G (giga) = ...... c (centi) = ......

M (mega) = ....... m (milli) = .......

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k (kilo) = ........ (micro) =........

2- Atmosphere of pressure, or 1 Atom ( = ............ kPa the SI unit)

The SI unit of is ..........

Absolute pressure is the sum of ............................. plus ..........................

...................... is the resistance to flow of a liquid .

The elastic modulus is defined as .........................

3- The dynamic viscosity of air at 15oC is 1.78 × 10-5 N.s/m2. Using Sutherland’s

equation, find the viscosity at 200oC .

4- What happens if the viscosity is too low ?

5- What happens if the viscosity is too high?

6- Explain why a water drop has surface tension.

7- The atmospheric pressure at the top and the bottom of a building are read by a

barometer to be 96 kPa and 98 kPa. If the density of air is 1.0 kg/m3 . What is the

height of the building? [204 m]

8- Determine the atmospheric pressure at a location where the barometric reading

is 750 mm Hg. Take the density of mercury to be 13600 kg/m3. [99.96 kPa]

9- The absolute pressure in water at a depth of 5 m is read to be 145 kPa.

Determine (a)the local atmospheric pressure , and (b) the absolute pressure at a

depth of 5 m in liquid whose specific gravity is 0.85 at the same location.

[96 kPa, 137.65 kPa] .

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