Archimedes’Principle

Abodyfullyorpartlysubmergedexperiencesabuoyantforceequaltotheweightoffluidthatitdisplaces.

VideoofArchimedes’Principle(from2:00):http://youtu.be/eQsmq3Hu9HA

Buoyancyinair

ArchimedesofSyracuse

287-212BCE

‘givemealeverandaplacetostandandIcanmovetheearth”

Inthebath(manchester)

Archimedes’principle: fullysubmerged

Considerafullysubmergeddiver

Thenetupwardforceonthediveris

Sothediverrises,staysstill,orsinksdependingonwhetherheraverage densityisless,equalto,orsmallerthanthewaterdensity.

Vwater displaced = Vdiver

Fb = mwater displaced g

= ⇢waterVdiver g

W = mdiver g

= ⇢diverVdiver g

Fnet = Fb �W

= (⇢water � ⇢diver)Vdiver g

Cartesiandivers…

Swimbladder exampleFishcontroltheirdepthbyadjustingthevolumeof"swimbladders.”Withitsswimbladdercollapsed,amackerelhasadensityof1.08g/cm3.Bywhatfractionofitsoriginalbodyvolumemustthefishinflateitsairsacstoreduceitsdensitytothatofwater?

Answer:weneedtoreducethedensityby1.08x,i.e.ρ1/ρ2 =1.08.

Sincevolumeisinverselyproportionaltodensity,V2/V1 =ρ1/ρ2 =1.08.

Sothenewvolumeneedstobe1.08xlarger,i.e.theswimbladderneedstobeinflatedto8%ofthefish’svolume

Sharksdon’thaveaswimbladder:doyouknowwhytheydon’tsink?

©JanHo

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Galilean thermometer

Buoyancyofspheresdependsonrho_w – rho_s (

(rho_w – rho_s)+ve ->floating;negative->floats

Now, water’sthermalexpansioncoefficientismuchlargerthanglass– andexpansionmeansreductionindensity,so..

(ρw – ρs) goesdownastemperaturegoesup

Therefore…spheres tendtosinkathighertemperature

Challenge:Nospherecaneverhaveexactlythesamedensityaswater.Sowhyistherealwaysaspherenearhalfway?

Archimedes’principle:floating

Theduckispartiallysubmerged.

Sothevolumeofdisplacedwaterissomefractionoftheduck’svolume

Also:duckisinequilibrium,sobuoyancyforceisbalancedbyweight:

Thefractionoftheduck’svolumethat’ssubmergedis:

Fb = W

) mwater displaced g = mduck g

) mwater displaced = mduck

Vwater displaced

Vduck=

mwater displaced/�watermduck/�duck

=�duck�water

Fb

W

Whydoesasteelboatfloat?

Itdisplacesalargervolumewaterthanthevolumeofmetalinit.

A200-tonshipentersthelockofacanal.Thefitbetweenthesidesofthelockandtheshipistightsothattheweightofthewaterleftinthelockafteritclosesismuchlessthan200tons.Cantheshipstillfloatifthequantityofwaterleftinthelockismuchlessthantheship�sweight?

PauseforthoughtArchimedes’principle