Static FluidsStatic Fluids

• Fluids are substances, such as liquids Fluids are substances, such as liquids and gases, that have no rigidity. A and gases, that have no rigidity. A fluid lacks a fixed shape and fluid lacks a fixed shape and assumes the shape of its container.assumes the shape of its container.

• In the liquid state, molecules can In the liquid state, molecules can flow; they freely move from position flow; they freely move from position to position by sliding over one to position by sliding over one another.another.

PressurePressure

• The pressure P acting on a fluid is the The pressure P acting on a fluid is the force exerted perpendicularly per unit of force exerted perpendicularly per unit of the fluid’s surface areathe fluid’s surface area

• Unit of pressure is the N/mUnit of pressure is the N/m22 or Pascal; or Pascal; 1 N/m1 N/m22 = 1 Pa (Pascal). = 1 Pa (Pascal).

• Atmospheric pressure at sea level is Atmospheric pressure at sea level is 1 atmosphere (atm) = 1.013 x 101 atmosphere (atm) = 1.013 x 1055 Pa. Pa.

• 1 atm = 14.7 lb/in1 atm = 14.7 lb/in22..

AF

P

Pressure in a Liquid.Pressure in a Liquid.

• A liquid in a container exerts forces A liquid in a container exerts forces against the walls and bottom of the against the walls and bottom of the container.container.

• For a liquid in a container, the For a liquid in a container, the pressure the liquid exerts against pressure the liquid exerts against the bottom of the container is the the bottom of the container is the weight of the liquid divided by the weight of the liquid divided by the area of the container bottom.area of the container bottom.

Measuring PressureMeasuring Pressure

• A manometer is a U-shaped A manometer is a U-shaped tube that is partially filled with tube that is partially filled with liquid.liquid.

• Both ends of the tube are open Both ends of the tube are open to the atmosphere.to the atmosphere.

• A container of gas is A container of gas is connected to one end of the U-connected to one end of the U-tube.tube.

• If there is a pressure If there is a pressure difference between the gas difference between the gas and the atmosphere, a force and the atmosphere, a force will be exerted on the fluid in will be exerted on the fluid in the U-tube. This changes the the U-tube. This changes the equilibrium position of the equilibrium position of the fluid in the tube.fluid in the tube.

AlsoAlso

atmc PP At point CAt point C

B'B PP The pressure at point B is the pressure The pressure at point B is the pressure of the gas.of the gas.

PPgaugegauge easily remembered as “hot dog” easily remembered as “hot dog”

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atm

gauge

B B C

B C B

let d h

P P P h D g

P P P P h D g

P h D g

From the figure:From the figure:

A BarometerA Barometer The atmosphere pushes on the The atmosphere pushes on the container of mercury which container of mercury which forces mercury up the closed, forces mercury up the closed, inverted tube. The distance d inverted tube. The distance d is called the is called the barometric barometric pressurepressure..From the figure:From the figure: atmBA PPP

andand

A

let d h

P h D g

Atmospheric pressure is equivalent to a column of mercury Atmospheric pressure is equivalent to a column of mercury 76.0 cm tall.76.0 cm tall.

DensityDensity• Density D of a substance is its mass per Density D of a substance is its mass per

unit volumeunit volume;;

• Objects composed of the same substance, Objects composed of the same substance, whatever the size or mass, have the same whatever the size or mass, have the same density under the same conditions of density under the same conditions of temperature and pressure.temperature and pressure.

• Temperature and pressure affect the Temperature and pressure affect the density of substances, appreciably for density of substances, appreciably for gases, but only slightly for liquids and gases, but only slightly for liquids and solids.solids.

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D

DensityDensity•How much a liquid weighs and how

much pressure it exerts depends on its density. – For the same depth, a denser liquid exerts

a greater pressure than a less dense liquid. – For liquids of the same density, the

pressure will be greater at the bottom of the deeper liquid.

• To convert a density in g/cm3 to kg/m3, multiply by 1000.

Columnar Fluid PressureColumnar Fluid Pressure

(sometimes called gauge pressure)(sometimes called gauge pressure) • pressure due to a column pressure due to a column

of fluid of height h and of fluid of height h and mass density D; mass density D;

• The pressure of a liquid at The pressure of a liquid at rest depends on the rest depends on the density and depth of the density and depth of the liquid.liquid.

• Liquids are practically Liquids are practically incompressible, so except incompressible, so except for changes in the for changes in the temperature, the density temperature, the density of a liquid is normally the of a liquid is normally the same at all depths.same at all depths.

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Columnar Fluid PressureColumnar Fluid Pressure• At a given depth, a given liquid At a given depth, a given liquid

exerts the same pressure exerts the same pressure against any surface - the against any surface - the bottom or sides of its container, bottom or sides of its container, or even the surface of an object or even the surface of an object submerged in the liquid to that submerged in the liquid to that depth.depth.

• Pressure a liquid exerts Pressure a liquid exerts depends only on its density and depends only on its density and depth. depth.

• Total pressure (or absolute Total pressure (or absolute pressure) Ppressure) Pabsoluteabsolute on a on a submerged surface equals the submerged surface equals the pressure the liquid exerts plus pressure the liquid exerts plus the atmospheric pressure Pthe atmospheric pressure Po o ((1 1

atm = 1.013 x 10atm = 1.013 x 1055 Pa) Pa) . . ( )absolute oP P h D g peanut hot dog

Fluid PressureFluid Pressure• Pressure of a liquid does not Pressure of a liquid does not

depend on the amount of depend on the amount of liquid.liquid.

• Neither the volume or total Neither the volume or total weight of the liquid matters.weight of the liquid matters.

• If you sampled water pressure If you sampled water pressure at 1 m beneath a large lake at 1 m beneath a large lake surface and 1 m beneath a surface and 1 m beneath a small pool surface, the small pool surface, the pressure would be the same.pressure would be the same.

• The fact that water pressure The fact that water pressure depends on depth and not on depends on depth and not on volume is illustrated by Pascal volume is illustrated by Pascal vases.vases.

• Water surface in each of the Water surface in each of the connected vases is at the connected vases is at the same level.same level.

• Occurs because the pressures Occurs because the pressures at equal depths beneath the at equal depths beneath the surface are the same.surface are the same.

Fluid Fluid PressurePressure

• At any point within a liquid, the forces that produce pressure are exerted equally in all directions.

• Pressure increases vertically downward.

• Pressure constant horizontally.

Forces Exerted By a FluidForces Exerted By a Fluid

• When the liquid is pressing When the liquid is pressing against a surface there is a against a surface there is a net force directed net force directed perpendicular to the perpendicular to the surface.surface.

• If there is a hole in the If there is a hole in the surface, the liquid initiallysurface, the liquid initially moves perpendicular to moves perpendicular to the surface.the surface.

• At greater depths, the net At greater depths, the net force is greater and the force is greater and the horizontal velocity of the horizontal velocity of the escaping liquid is greater.escaping liquid is greater.

Transmission of Pressure: Transmission of Pressure: Pascal’s Principle.Pascal’s Principle.

• Pascal’s Principle: A CHANGE IN Pascal’s Principle: A CHANGE IN PRESSURE IN A CONFINED FLUID PRESSURE IN A CONFINED FLUID IS TRANSMITTED WITHOUT IS TRANSMITTED WITHOUT CHANGE TO ALL POINTS IN THE CHANGE TO ALL POINTS IN THE FLUID.FLUID.

• Ex. Ex. Hydraulic lift.Hydraulic lift. • Hydraulic piston apparatus uses Hydraulic piston apparatus uses

an incompressible fluid to an incompressible fluid to transmit pressure from a small transmit pressure from a small cylinder to a large cylinder.cylinder to a large cylinder.

• According to Pascal’s Principle, According to Pascal’s Principle, the pressure in the small cylinder the pressure in the small cylinder resulting from the application of resulting from the application of FF11 to a frictionless piston is to a frictionless piston is transmitted undiminished to the transmitted undiminished to the larger piston.larger piston.

Transmission of pressure: Transmission of pressure: Pascal’s Principle.Pascal’s Principle.

PP11 = P = P22

• AA22 is larger than A is larger than A11, so the , so the

force exerted by the large force exerted by the large piston is greater than the piston is greater than the force exerted on the small force exerted on the small piston. piston.

• AMA (actual mechanical AMA (actual mechanical advantage) for hydraulic lift: advantage) for hydraulic lift:

2

2

1

1

A

F

AF

1

2

F

FAMA

Apply a force F1 here to a piston of cross-sectional area A1.

The applied force is transmitted to the piston of cross-sectional area A2 here.

F2 F1A2

A1

Transmission of pressure: Transmission of pressure: Pascal’s Principle.Pascal’s Principle.

• The figure shows a hydraulic The figure shows a hydraulic system used with brakes. The system used with brakes. The force F is applied force F is applied perpendicularly to the brake perpendicularly to the brake pedal. The brake pedal rotates pedal. The brake pedal rotates about the axis shown in the about the axis shown in the drawing and causes a force to drawing and causes a force to be applied perpendicularly to be applied perpendicularly to the input piston in the master the input piston in the master cylinder. The resulting cylinder. The resulting pressure is transmitted by the pressure is transmitted by the brake fluid to the output brake fluid to the output plungers which are covered plungers which are covered with the brake linings. The with the brake linings. The linings are pressed against linings are pressed against both sides of a disc attached to both sides of a disc attached to the rotating wheel.the rotating wheel.

BuoyancyBuoyancy • Buoyancy: the apparent loss Buoyancy: the apparent loss

of weight of objects when of weight of objects when submerged in a liquid.submerged in a liquid.

• Easier to lift objects under Easier to lift objects under water surface than to lift it water surface than to lift it above the water surface.above the water surface.

• When submerged, water When submerged, water exerts an upward force that exerts an upward force that is opposite in direction to is opposite in direction to gravity. Upward force called gravity. Upward force called the the buoyant forcebuoyant force..

BuoyancyBuoyancy • Forces exerted by liquid Forces exerted by liquid produce pressure against produce pressure against the submerged object.the submerged object.

• Forces are greater at Forces are greater at greater depths; forces are greater depths; forces are equal at the same depth on equal at the same depth on opposite sides of the object.opposite sides of the object.

• Forces acting upward on the Forces acting upward on the bottom of the object greater bottom of the object greater than those acting downward than those acting downward on top of the object, simply on top of the object, simply because the bottom of the because the bottom of the object is deeper.object is deeper.

• Difference in upward and Difference in upward and downward forces is the downward forces is the buoyant force, B.buoyant force, B.

• Fs refers to the force the Fs refers to the force the scale exerts on the mass m. scale exerts on the mass m. You may also refer to Fs as You may also refer to Fs as the tension in a supporting the tension in a supporting string.string.

BuoyancyBuoyancy• If the weight of the If the weight of the

object is greater than the object is greater than the buoyant force, the object buoyant force, the object will sink (as in figure a).will sink (as in figure a).

• If the weight of the If the weight of the object is equal to the object is equal to the buoyant force, the net buoyant force, the net force on the object is force on the object is zero and the submerged zero and the submerged object will remain at any object will remain at any level (as in figure b).level (as in figure b).

• If the weight of the If the weight of the object is less than the object is less than the buoyant force, the object buoyant force, the object will rise to the surface will rise to the surface and float.and float.

BuoyancyBuoyancy

• When an object is submerged in a liquid, When an object is submerged in a liquid, the liquid level will rise.the liquid level will rise.

• Liquid is displaced or moved elsewhere.Liquid is displaced or moved elsewhere.

• The volume of the liquid displaced is equal The volume of the liquid displaced is equal to the volume of the submerged object.to the volume of the submerged object.

• A completely submerged object always A completely submerged object always displaces a volume of liquid equal to its displaces a volume of liquid equal to its own volume.own volume.

Archimede’s PrincipleArchimede’s Principle

• When an object is immersed in a fluid, it When an object is immersed in a fluid, it appears to weigh less.appears to weigh less.

• Archimede’s Principle: THE BUOYANT Archimede’s Principle: THE BUOYANT FORCE EXERTED ON A BODY WHOLLY OR FORCE EXERTED ON A BODY WHOLLY OR PARTLY IMMERSED IN A FLUID IS EQUAL PARTLY IMMERSED IN A FLUID IS EQUAL TO THE WEIGHT OF THE FLUID DISPLACED TO THE WEIGHT OF THE FLUID DISPLACED BY THE BODY.BY THE BODY.

• Archimede’s Principle applies to both Archimede’s Principle applies to both liquids and gases, which are fluids.liquids and gases, which are fluids.

• ImmersedImmersed refers to either completely or refers to either completely or partially submerged.partially submerged.

Archimede’s PrincipleArchimede’s Principle• Buoyant force: upward force on the object when it is Buoyant force: upward force on the object when it is

immersed in water. immersed in water.

• The buoyant force (BF) is the weight of the displaced The buoyant force (BF) is the weight of the displaced fluid - not the weight of the submerged object.fluid - not the weight of the submerged object.

• BF = (mass in air - mass in fluid)·gBF = (mass in air - mass in fluid)·gravityravity

• BF = weight in air - weight in fluidBF = weight in air - weight in fluid

• BF = weight of displaced fluid (DBF = weight of displaced fluid (Dfluidfluid·V·Vobject submergedobject submerged·g)·g)

• Apparent weight of a submerged object is its weight in Apparent weight of a submerged object is its weight in air minus the buoyant force.air minus the buoyant force.

Apparent weight = Apparent weight = m·g – BF = m·g - Dm·g – BF = m·g - Dfluidfluid·V·Vobject submergedobject submerged·g·g • For an object that is floating or is submerged but not

sinking: m mobjectobject·g = D·g = Dfluidfluid·V·Vobject submergedobject submerged·g·g

Flotation Flotation

• Principle of Flotation: A FLOATING Principle of Flotation: A FLOATING OBJECT DISPLACES A WEIGHT OF OBJECT DISPLACES A WEIGHT OF FLUID EQUAL TO ITS OWN WEIGHT.FLUID EQUAL TO ITS OWN WEIGHT.

• A simple relationship between the A simple relationship between the weight of a submerged object and weight of a submerged object and the buoyant force can be found by the buoyant force can be found by considering their ratio:considering their ratio:

fluidDobjectD

BFobjectFw

FlotationFlotation

• Shipbuilding and the principle of Shipbuilding and the principle of flotation:flotation: – A solid 1-ton block of iron is nearly 8 A solid 1-ton block of iron is nearly 8

times as dense as water, so when it is times as dense as water, so when it is submerged, it will displace 1/8 ton of submerged, it will displace 1/8 ton of water (not an amount equal to 8 tons of water (not an amount equal to 8 tons of water).water).

– Reshape the iron block into a bowl and Reshape the iron block into a bowl and submerge, displaces a greater volume of submerge, displaces a greater volume of water. The deeper the bowl is immersed water. The deeper the bowl is immersed at the surface, the more water is at the surface, the more water is displaced and the greater is the buoyant displaced and the greater is the buoyant force exerted on the bowl.force exerted on the bowl.

FlotationFlotation• When the weight of the displaced water When the weight of the displaced water

equals the weight of the bowl - flotation. equals the weight of the bowl - flotation. Flotation occurs when the weight of the Flotation occurs when the weight of the bowl equals the buoyant force.bowl equals the buoyant force.

• Every ship must be designed to displace Every ship must be designed to displace a weight of water equal to its own a weight of water equal to its own weight.weight.

• Submarines:Submarines: – Displace a weight of water equal to Displace a weight of water equal to

its own weight, it remains at a its own weight, it remains at a constant depth.constant depth.

– Displaces a weight of water greater Displaces a weight of water greater than its own weight, rises.than its own weight, rises.

– Displaces a weight of water less than Displaces a weight of water less than its own weight, sinks.its own weight, sinks.

Buoyancy in Two Liquids of Buoyancy in Two Liquids of Differing DensityDiffering Density

• If you have an object submerged in two liquids of If you have an object submerged in two liquids of different density, such that the upper portion of different density, such that the upper portion of the object is located in the upper liquid and the the object is located in the upper liquid and the lower portion of the object is located in the lower lower portion of the object is located in the lower liquid, the total buoyant force on the object is liquid, the total buoyant force on the object is equal to the weight of the object equal to the weight of the object

BF = DBF = Dfluidfluid·V·Vobject submergedobject submerged·g·g

• Ex. A piece of wood floating partially in water Ex. A piece of wood floating partially in water and partially in oil. The density of the oil is less and partially in oil. The density of the oil is less than the density of the water. than the density of the water.

• Gravity cancels out.Gravity cancels out.

gVDBFBF objectobjectwateroil

gVD)gVD()gVD( objectobjectwateroil

objectobjectwateroil VD)VD()VD(

( ) ( )oil water object object

V A h

D V D V D V

• If d is the height of the object (the 4 cm in the figure), let y = the portion of the object in the more dense liquid and d-y = the portion of the object in the less dense liquid.

• This equation can then be solved for the unknown variable.

( ) ( )oil water object objectD A d y D A y D V

Pressure ExamplePressure Example• Water is to be pumped to the top of the Empire Water is to be pumped to the top of the Empire

State Building, which is 366 m high. What State Building, which is 366 m high. What gauge pressure is needed to raise the water to a gauge pressure is needed to raise the water to a height of 366 m? The density of water is 1000 height of 366 m? The density of water is 1000 kg/mkg/m33..

Pa3586800m

N3586800P

s

m9.8

m

kg1000m366P

gDhP

2

23

Buoyant ForceBuoyant Force

• An object weighing An object weighing 300 N in air is 300 N in air is immersed in water immersed in water after being tied to a after being tied to a string connected to a string connected to a balance. The scale balance. The scale now reads 265 N. now reads 265 N. Immersed in oil, the Immersed in oil, the object appears to object appears to weigh 275 N.weigh 275 N.

• A. Find the density of A. Find the density of the object.the object.

333

2

air

air

33

3

OHair

m

kg8571.5

m10x3.5714

kg30.61D

kg30.61

s

m9.8

N300g

Fwm

gmFw

m10x3.5714V

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kg1000

N35gD

BFV

gVDBF

N35N265N300BF

FwFwBF2

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D

s

m2

89

Buoyant ForceBuoyant Force

• B. Determine the density of the oil.B. Determine the density of the oil.

3

233

33

oilair

m

kg.

s

m9.8m10x3.5714

N25gV

BF

m10x3.5714V

displacedfluidofvolumeobjectofvolume

gVDBF

N25N275N300BF

FwFwBF

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