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Eureka!. Specific Gravity – The REST of the story…. AKA “Owl” Archimedes saved his life by taking a bath!. Hiero of Syracuse (really!)…. challenged Archimedes. to determine if his new crown was made of pure gold.

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specific gravity the rest of the story

Eureka!

Specific Gravity – The REST of the story…

AKA

“Owl” Archimedes saved his life by taking a bath!

hiero of syracuse really

Hiero of Syracuse (really!)…

challenged Archimedes

to determine if his new crown was made of pure gold.

therefore

Fg

F?

FT

Therefore…

F?

FT

+

=

Fg

this new force is called

Fg

F?

FT

This new force is called…

Oh, BOY!

F?

FT

+

=

Fg

the buoyant force

Fg

F?

FT

the BUOYANT force!

Oh, BOY!

F?

FT

+

=

Fg

the buoyant force1

Fg

FB

FT

the BUOYANT force!

Oh, BOY!

FB

FT

+

=

Fg

because h 2 h 1 the force up is greater than the force down

FB = |Fup| - |Fdown|

Because h2 > h1, the force up is greater than the force down.

= PbotAbot - PtopAtop

= ρgh2Abot – ρgh1Atop

{ Abot = Atop }

= ρgA(h)

h1

= ρgV

h2

h = h2 - h1

since the fluid was at rest

|FB|= Fgfluid

Since the fluid was at rest…

= ρfluidgV

= mfluidg

Remember that:

h1

{ ρ = m/V }

h2

h = h2 - h1

{ m = ρV }

so f b f g fluid

So | FB |= | Fgfluid |

Oh, BOY!

h1

h2

h = h2 - h1

so f b f g fluid1

Is this

always true?

So | FB |= | Fgfluid |

h1

h2

h = h2 - h1

let s consider a still fluid

Let’s consider a still fluid:

The fluid was there already, and if we encase it in a plastic bag that has the same density as the fluid, it will continue to be there.

f b f g fluid

| FB |= | Fgfluid |

Oh, BOY!

So the buoyant force on an object equals the weight of the fluid displaced, regardless of the shape of the object.

slide24

If Fnet is zero, then the object is neutrally buoyant and it will neither rise nor sink while it is under the surface.

Fnet = Fgobject + FB

FB

Fnet = 0 N

Fg

slide26

float

(i.e. The object would displace enough fluid such that Fnet = ________.

Please note also, if Fnetwere positive, the object would simply __________ if it were not held below the surface.

0 N

remember that sg o f

Fg

FB

FT

Oh, yeah...

Remember that SG = ρo/ρf

FB

FT

+

=

Fg

so sg o f

SG = mo/Vo/mf/Vf

SG = mo/mf

So… SG = ρo/ρf

SG = mog/mfg

SG = Fgo/Fgf

SG = Fgo/|FB|

SG = Fgo/ΔFgo

slide30

Eureka!

So… Archimedes realized that an object’s specific gravity (relative to the fluid used) equaled the weight of the object out of the water divided by the change of weight that occurred when the object was submerged in the fluid!

Oh, BOY!

SG = Fg/ΔFg

slide31

SG = Fgo/ΔFgo

SG = Fgo/|Fgo– Fgo’|

SG = ρoVo/ρfVf

Note:

SG = ρo/ρf

If the fluid involved is water at 4ºC, then this ratio is the specific gravity for the object, not just a relative SG.

for floating objects objects sink enough that

|FB| = |Fg|

ρo = 0.2 g/mL

For floating objects, objects sink enough that:

ρf.Vdisplaced.g=ρo.Vo.g

Vdisplaced/Vo = ρo/ρf

So 80% of the ball is above the water.

hydrometers follow

ρ1/ρ2 = V2/V1

Hydrometers follow:

ρ1/ρ2 = A2h2/A1h1

ρ1/ρ2 = h2/h1

h2/h1

Ex. .8/1= .8 /1

If placed into a denser fluid: .8/2= x /1

hydrometers follow1

ρ1/ρ2 = V2/V1

Hydrometers follow:

x = 0.4

ρ1/ρ2 = A2h2/A1h1

ρ1/ρ2 = h2/h1

Ex. .8/1= .8 /1

.8/2= x /1

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