Fluids and elasticity
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Fluids and Elasticity. Chapter 15. Density ( r ). r = mass/volume Rho ( r ) – Greek letter for density Units - kg/m 3 Specific Gravity = Density of substance Density of water (4 o C) Unitless ratio Ex: Lead has a sp. Gravity of 11.3 (11.3 times denser than water. Ex: 1.

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Density r
Density (r)

r = mass/volume

  • Rho ( r) – Greek letter for density

  • Units - kg/m3

  • Specific Gravity = Density of substance

    Density of water (4oC)

  • Unitless ratio

  • Ex: Lead has a sp. Gravity of 11.3 (11.3 times denser than water


Ex: 1

Estimate the mass of air in this classroom


Pressure
Pressure

  • Force per unit area

  • P = F/A

  • Unit - N/m2 (Pascal)

  • The larger the area, the less the pressure

    • Shoeshoes

    • Elephant feet

    • Bed of nails


Fluid pressure
Fluid Pressure

  • A fluid exerts the same pressure in all directions at a given depth

  • P = Po + rgh

  • The atmosphere is a fluid

  • Po often 1 atm (101.3 kPa)


Pressure example 1
Pressure: Example 1

A water storage tank is 30 m above the water faucet in a house. Calculate the pressure at the faucet:

We will neglect the atmospheric pressure since it is the same at the tank and at the surface

DP = rgh = (1000 kg/m3)(9.8 m/s2)(30 m)

DP = 29,000 kgm2/m3s2 = 29,000 kg m/s2m2

DP = 29,000 N/m2


Pressure example 2
Pressure: Example 2

The Kraken can live at a depth of 200 m. Calculate the pressure the creature can withstand (neglect atmospheric pressure)


Pressure example 21
Pressure: Example 2

DP = rgh = (1000 kg/m3)(g)(200 m)

DP = 1.96 X 106 N/m2


Atmospheric pressure
Atmospheric Pressure

  • 1 atm = 1.013 X 105 N/m2 = 101.3 kPa

  • 1 bar = 1 X 105 N/m2 (used by meteorologists)

  • Gauge pressure

    P = Patm + PG

    Absolute pressure atmospheric pressure Gauge pressure



Straw example
Straw Example

You can pick up soda in a straw using your finger. Why doesn’t the soda fall out?


Another straw example
Another Straw Example

What pushes soda up a straw when you drink through it?


Torrecilli s work
Torrecilli’s Work

P = Patm + PG

P = Patm + rgh

What is the highest column of water that the atmosphere can support?


P = Patm + rgh

0 = 1.013 X 105 N/m2 + (1000kg/m3)(9.8m/s2)(h)

h = 10.3 m

  • No vacuum pump can pump more than ~30 feet


Try the same calculation with mercury

P = Patm + rgh

0 = 1.013 X 105 N/m2 + (13,600kg/m3)(9.8m/s2)(h)

h = 0.760 m (760 mm)

1 atm = 760 mm Hg (760 torr)


Can an astronaut attach suction cups to the boots of his spacesuit to help him climb around the space shuttle while in space?


Pascal s principle
Pascal’s Principle spacesuit to help him climb around the space shuttle while in space?

  • Pressure applied to a confined fluid increases the pressure the same throughout

    Pin = Pout

    Fin = Fout

    Ain Aout

    for pistons:

    DF = rg(A1 + A2)d2


Pascal s principle ex 1
Pascal’s Principle: Ex 1 spacesuit to help him climb around the space shuttle while in space?

A hydraulic lift can produce 200 lb of force. How heavy a car can be lifted if the area of the lift is 20 times larger that the input of the Force?

Fin = Fout

Ain Aout

Fout = Fin Aout = (200 lb) (20) = 4000 lbs

Ain 1


Pascal s principle ex 11
Pascal’s Principle: Ex 1 spacesuit to help him climb around the space shuttle while in space?

A hydraulic lift has the car rest ona 25 cm pipe. The lift the car, compressed air pushes on a 6.0 cm pipe.

  • Calculate the force needed to lift a 1300 kg car.

  • Calculate how much the air pressure force must be increased to lift the car 2.0 m.


Buoyancy
Buoyancy spacesuit to help him climb around the space shuttle while in space?

  • Buoyancy

    • The “lift” provided by water

    • Objects weight less in water than out

  • Caused by pressure differential between top and bottom of an object.

    Fbouyant = rgV


Derivation of the buoyancy formula
Derivation of the Buoyancy Formula spacesuit to help him climb around the space shuttle while in space?

Fb = F2 – F1

P = F/A

F = PA

F = rghA

Fb = rgh2A – rgh1A

Fb = rgA(h2 -h1)

Fb = rgV


Archimedes principle
Archimedes Principle spacesuit to help him climb around the space shuttle while in space?

“The bouyant force on an object equals the weight of fluid displaced by the object”

w’ = weight of an object in water (or any liquid)

w’ = mg - Fb


Buoyancy example 1
Buoyancy: Example 1 spacesuit to help him climb around the space shuttle while in space?

A 7000-kg ancient statue lies at the bottom of the sea. Its volume is 3.0 m3. How much force is needed to lift it?

Fb = rgV

Fb = (1000 kg/m3)(9.8 m/s2)(3.0m3)

Fb = 2.94 X 104 kg-m/s2

Fb = 2.94 X 104 N

Fb

mg


w’ = mg - F spacesuit to help him climb around the space shuttle while in space?b

w’ = (7000 kg)(9.8m/s2) - 2.94 X 104 N

w’ = 3.92 X 104 N

Say, isn’t w’ just the sum of the forces?

Yep.

SF = w’

Fb

mg


Buoyancy example 2
Buoyancy: Example 2 spacesuit to help him climb around the space shuttle while in space?

Archimedes tested a crown for the king. Out of water, it masses 14.7 kg. In water, it massed 13.4 kg. Was the crown gold?

w’ = mcrg – Fb

w’ = mcrg – rgVcr

(13.4 kg)(g) = (14.7 kg)(g) – (1000 kg/m3)(g)(Vcr)

131 N = 144 N – (9800 kg/ms2)(Vcr)

Vcr = 0.00133 m3


Now we can calculate the density of the crown: spacesuit to help him climb around the space shuttle while in space?

  • = m/V = 14.7 kg/0.00133 m3

  • = 11,053 kg/m3

    Gold’s density is about 19,000kg/m3. This is much closer to lead.


Example 3
Example 3 spacesuit to help him climb around the space shuttle while in space?

A cube of wood that is 10 cm on a side is held underwater by tying a string to the cube and the bottom on the container. The wood has a density of 700 kg/m3.

  • Draw a free body diagram showing all the forces on the block.

  • Calculate the force of bouyancy

  • Calculate the tension in the string.


Floating
Floating spacesuit to help him climb around the space shuttle while in space?

  • Objects that are less dense than water will float

  • Part of the object will be above the water line

  • A case of static equilibrium

    SF = 0

Fb

mg


Floating example 1
Floating: Example 1 spacesuit to help him climb around the space shuttle while in space?

A 1200 kg log is floating in water. What volume of the log is under water?

SF = 0

SF = 0 = mg – Fb

mg = Fb

mg = rgVlog

Vlog = mg

rg

Fb

mg


V spacesuit to help him climb around the space shuttle while in space?log = m (Hey, the g’s cancel!)

r

Vlog = 1200 kg = 1.2 m3

1000 kg/m3


Floating example 11
Floating: Example 1 spacesuit to help him climb around the space shuttle while in space?

A wooden raft has a density of 600 kg/m3, an area of 5.7 m2, and a volume of 0.60 m3. How much of the raft is below water in a freshwater lake?


Let’s first calculate the mass of the raft: spacesuit to help him climb around the space shuttle while in space?

r = m/V

m = rV = (600 kg/m3)(0.60 m3) = 360 kg

Now we can worry about the raft.

SF = 0

SF = 0 = mg – Fb

mg = Fb

mg = rgVsubmerged

mg = rghsubmergedA


mg = spacesuit to help him climb around the space shuttle while in space?rghsubmergedA

m = rhsubmergedA (Hey, the g’s cancelled!)

hsubmerged = m/rA

hsubmerged = 360 kg = 0.063 m

(1000 kg/m3)(5.7 m2)


Floating example 3
Floating: Example 3 spacesuit to help him climb around the space shuttle while in space?

Suppose a continent is floating on the mantle rock. Estimate the height of the continent above the mantle (assume the continent is 35 km thick).


S spacesuit to help him climb around the space shuttle while in space?F = 0 = mg – Fb

0 = mcg – rmangVc(submerged)

mcg = rmangVc(submerged)

mc = rmanVc(submerged)

We don’t know the mass of the continent

rc = mc/Vc(total)

mc = rcVc(total)

mc = rmanVc(submerged)


m spacesuit to help him climb around the space shuttle while in space?c = rmanVc(submerged)

mc = rcVc(total)

rmanVc(submerged) = rcVc(total)

Vc(submerged) = rc = (2800 kg/m3) = 0.85

Vc(total) rman (3300 kg/m3)

This means that 85% of the continent is submerged, and only 15% is above:

(0.15)(35 km) = 5.25 km


Floating ex 4
Floating: Ex 4 spacesuit to help him climb around the space shuttle while in space?

A block is placed in water and 5.8 cm is submerged. The same block is placed in an unknown liquid and 4.6 cm is submerged. Calculate the density of the unknown liquid. Assume the same face of the block pointed downward in both cases (A).


Fluid flow
Fluid Flow spacesuit to help him climb around the space shuttle while in space?



Equation of continuity
Equation of Continuity “in layers”)

A1v1 = A2v2

A = Area of a pipe

v = velocity of the liquid


Equation of continuity1
Equation of Continuity “in layers”)

v1A1 = v2A2

  • Fluid will flow faster through a smaller opening

  • Placing your finger over a hose opening.


The term “vA” is the “volume rate of flow” “in layers”)

A = m2

v = m/s

vA = m3/s

Q = vA


Eqn of continuity example 1
Eqn. Of Continuity: Example 1 “in layers”)

A garden hose has a radius of 1.00 cm and the water flows at a speed of 0.80 m/s. What will be the velocity if you place your finger over the hose and narrow the radius to 0.10 cm?

A1 = pr2 = (3.14)(0.01 m)2 = 3.14 X 10-4 m2

A2 = pr2 = (3.14)(0.001 m)2 = 3.14 X 10-6 m2


A “in layers”)1v1 = A2v2

v2 = A1v1

A2

v2 = (3.14 X 10-4 m2)(0.80 m/s) = 80 m/s

(3.14 X 10-6 m2)


Eqn of continuity example 2
Eqn. Of Continuity: Example 2 “in layers”)

A water hose 1.00 cm in radius fills a 20.0-liter bucket in one minute. What is the speed of water in the hose?

A1 = pr2 = (3.14)(1 cm)2 = 3.14 cm2

Remember that Av is volume rate of flow.

A2v2=20.0 L 1 min 1000 cm3= 333 cm3/s

1 min 60 s 1 L


A “in layers”)1v1 = A2v2

v1 = A2v2/A1

v1 = 333 cm3/s = 160 cm/s or 1.60 m/s

3.14 cm2


Eqn of continuity example 3
Eqn. Of Continuity: Example 3 “in layers”)

A sink has an area of about 0.25 m2. The drain has a diameter of 5 cm. If the sink drains at 0.03 m/s, how fast is water flowing down the drain?

Ad = pr2 = (p)(0.025 m)2 = 1.96 X 10-3 m3

Advd = Asvs

vd = Asvs/Ad=[(0.25 m2)(0.03 m/s)]/(1.96 X 10-3 m3)

vd = 3.82 m/s


Eqn of continuity example 4
Eqn. Of Continuity: Example 4 “in layers”)

The radius of the aorta is about 1.0 cm and blood passes through it at a speed of 30 cm/s. A typical capillary has a radius of about 4 X 10-4 cm and blood flows through it at a rate of 5 X 10-4 m/s. Estimate how many capillaries there are in the human body.


A “in layers”)ava = NAcvc (N is the number of capillaries)

Aa = pr2 = (3.14)(0.01 m)2 = 3.14 X 10-4 m2

Ac = pr2 = (3.14)(4 X 10-6 cm)2 = 5.0 X 10-11 cm2

N = Aava/ Acvc

N = (3.14 X 10-4 m2)(0.30 m/s) = ~ 4 billion

(5.0 X 10-11 cm2)(5 X 10-4 m/s)


Eqn of continuity example 5
Eqn. Of Continuity: Example 5 “in layers”)

How large must a heating duct be to replenish the air in a room 300 m3 every 15 minutes? Assume air moves through the vent at 3.0 m/s.


A “in layers”)dvd = Arvr

Arvr = volume rate of flow:

Arvr = 300 m3 1 min = 0.333 m3/s

15 min 60 s

Ad = Arvr/vd

Ad = 0.333 m3/s = 0.11 m2

3.0 m/s


Bernoulli s principle
Bernoulli’s Principle “in layers”)

The velocity and pressure of a fluid are inversely related.


Why does a shower curtain sometimes “attack” a person taking a shower?

What will happen to closed windows during a tornado? Will they blow in or out?


Applications of bernoulli s principle
Applications of Bernoulli’s Principle taking a shower?

1. Airplane wing


Applications of bernoulli s principle1
Applications of Bernoulli’s Principle taking a shower?

  • Prairie Dog Burrows

    • Air moves faster (lower pressure) at the top

    • Draws air through the burrow

    • The exact same thing happens with our chimneys


Applications of bernoulli s principle2
Applications of Bernoulli’s Principle taking a shower?

3. Spray Paint

Flow of air (low pressure)


Applications of bernoulli s principle3
Applications of Bernoulli’s Principle taking a shower?

4. Dime in a cup


Bernoulli s equation
Bernoulli’s Equation taking a shower?

Pt + ½rvt2 + rgyt = Pb + ½rvb2 + rgyb

(note: you often have to use the Eqn. Of Continuity in these situations:)

A1v1 = A2v2


Pipe from a water reservoir to a house

Pipe from a house into a sewer pipe


If there is no change in altitude, the equation simplifies: taking a shower?

Pt + ½rvt2 + rgyt = Pb + ½rvb2 + rgyb

Pt + ½rvt2 = Pb + ½rvb2


Bernoulli s equation example 1
Bernoulli’s Equation: Example 1 taking a shower?

A water heater in the basement of a house pumps water through a 4.0 cm pipe at 0.50 m/s and 3.0 atm. What will be the flow speed and pressure through a 2.6 cm spigot on the second floor, 5.0 m above?

3.0 atm 1.013 X 105 N/m2 = 3.0 X 105 N/m2

1 atm


Flow speed: taking a shower?

Atvt = Abvb

vt = Abvb/At (Remember A = pr2)

vt = pr2bvb(Hey, the p’s cancel!)

pr2t

vt = r2bvb= (0.02 m)2(0.50m/s) = 1.2 m/s

r2t (0.013 m)2


Now the pressure: taking a shower?

Pt + ½rvt2 + rgyt = Pb + ½rvb2 + rgyb

Pt + ½(1000)(1.2)2 + (1000)(9.8)(5) = 3.0X105 + ½(1000)(0.50)2 + (1000)(9.8)(0)

Pt = 2.5 X 105 N/m2


Bernoulli s equation example 2
Bernoulli’s Equation: Example 2 taking a shower?

A drunken redneck shoots a hole in the bottom of an aboveground swimming pool. The hole is 1.5 m from the top of the tank. Calculate the speed of the water as it comes out of the hole.

yt = 1.5 m

yb = 0 m


The top of the pool is a taking a shower?much larger area than the hole. We will assume that the vt = 0.

Pt + ½rvt2 + rgyt = Pb + ½rvb2 + rgyb

Pt + rgyt = Pb + ½rvb2 + rgyb

Also, both the top and the hole are open to the atmosphere, so Pt = Pb

Pt + rgyt = Pb + ½rvb2 + rgyb

rgyt = ½rvb2 + rgyb


Set the bottom of the pool as y taking a shower?b = 0.

rgyt = ½rvb2 + rgyb

rgyt = ½rvb2

vb2 = 2rgyt/r

vb2 = 2gyt

vb2 = (2)(9.8m/s2)(1.5 m)

vb = 5.42 m/s


Example 31
Example 3 taking a shower?

A hydroelectric dam is 200 m above the power plant. The inlet hose at the top has a diameter of 100 cm, and the outlet hose to the turbine has a diameter of 50 cm.

  • Calculate the speed of the water into the turbine (both are open to the atmosphere)


Elasticity hooke s law
Elasticity: Hooke’s Law taking a shower?

  • Hooke’s Law – usually used with a spring

  • Can consider anything to be like a spring

  • F = kDL (F=kxspring)

  • k = proportionality (spring) constant

  • Can’t stretch things forever


Elastic region – material will still bounce back taking a shower?

Plastic region – material will not return to original length (but has not broken)

F = kDL

This is only linear

in the proportional

region



Elastic region young s modulus e
Elastic Region: Young’s Modulus(E) taking a shower?

Stress = Force = F

Area A

Strain = Change in length = DL

Original length Lo


Y = taking a shower?stress

strain

Y = F/A or F = Y DL

DL/Lo A Lo


Young s modulus example 1
Young’s Modulus: Example 1 taking a shower?

A 1.60 m long steel piano wire has a diameter of 0.20 cm. How great is the tension in the wire if it stretches 0.30 cm when tightened?

A = pr2 = p(0.0010 m)2 = 3.1 X 10-6 m2

F = Y DL

A Lo

F = AY DL

Lo


F = AY taking a shower?DL

Lo

F = (3.1 X 10-6m2)(200 X 1011N/m2)(0.0030 m)

1.60 m

F = 1200 N


Young s modulus example 2
Young’s Modulus: Example 2 taking a shower?

A steel support rod of radius 9.5 mm and length 81 cm is stretched by a force of 6.2 X 104 N (about 7 tons).

  • Calculate the stress.

  • Calculate the elongation.


Area = taking a shower?pr2 = p(0.0095 m2

Stress = Force = 6.2 X 10-4m = 2.2 X 108 N/m2

Area 2.84 X 10-4 m2

F = YDL

A Lo

DL = FL = (6.2 X 104 N)(0.81m)

YA (200 X 109 N/m2)(2.84 X 10-4 m2)

DL = 8.84 X 10-4 m = 0.89 mm


Young s modulus example 21
Young’s Modulus: Example 2 taking a shower?

A 2.0 m long, 1.0 mm diameter wire is suspended. Hanging a 4.5 kg mass stretches the wire length by 1.00 mm.

  • Calculate Young’s modulus

  • Identify the material from the table.


The three types of stress
The Three Types of Stress taking a shower?

Stretching Squeezing Horizontal


Other modulus
Other Modulus’ taking a shower?

Shear Modulus – Used for shear stress

Bulk Modulus – Used for even compression on all sides (an object when submerged)


Fracture
Fracture taking a shower?

  • Breaking Point

  • Uses

    • Tensile Strength – Stretching

    • Compressive Strength – under a load

    • Shear Stress – Shearing

  • Safety Factor – reciprocal that is multiplied by the tensile strength

  • Ex: A safety factor of 3 means you will only use 1/3 of the maximum stress


Fracture example 1
Fracture: Example 1 taking a shower?

A concrete column 5 m tall will have to support 1.2 X 105 N (compression). What area must it have to have a safety factor of 6?

Max stress = (1/6)(compressive strength)

Max stress = (1/6)(20 X 106 N/m2)= 3.3X106 N/m2

Stress = F

A


Stress = taking a shower?F

A

Area = F = (1.2 X 105 N) = 3.64 X 10-2 m

Stress 3.3 X 106N/m2


How much will the column compress under the load? taking a shower?

F = EDL

A Lo

DL = FL = (1.2 X 105 N)(5 m)

EA (20 X 109 N/m2)(3.64 X 10-2 m2)

DL = 8.3 X 10-4 m = 0.83 mm


Fracture example 2
Fracture: Example 2 taking a shower?

Spider-man’s webbing has a tensile strength of 600 X 106 N/m2 and he wishes to use a safety factor of 3. What is the diameter of the webbing if the maximum force at the bottom of a swing is 1500 N?


Maximum Stress taking a shower?

(1/3)(600 X 106 N/m2) = 200 X 106 N/m2

Stress = Force

Area

Area = Force = 1500 N = 7.5 X 10-6m2

Stress 200X106N/m2

Area = pr2

r = (A/p)1/2 = 1.55 X 10-3 m or 1.55 mm

Diameter = 3.10 mm


Concrete
Concrete taking a shower?

  • Concrete is much stronger under compression than tension

    • Tensile Strength – 2 X 106N/m2

    • Compressive Strength – 20 X 106N/m2

  • Prestressed concrete – rods or mesh are stretched when the concrete is poured. Released after concrete dries.

  • Now under compression


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