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F B = w fl , size 12{F rSub { size 8{B} } =w rSub { size 8{"fl"} } } {}

where F B size 12{F rSub { size 8{B} } } {} is the buoyant force and w fl size 12{w rSub { size 8{"fl"} } } {} is the weight of the fluid displaced by the object. Archimedes’ principle is valid in general, for any object in any fluid, whether partially or totally submerged.

Archimedes’ principle

According to this principle the buoyant force on an object equals the weight of the fluid it displaces. In equation form, Archimedes’ principle is

F B = w fl , size 12{F rSub { size 8{B} } =w rSub { size 8{"fl"} } } {}

where F B size 12{F rSub { size 8{B} } } {} is the buoyant force and w fl size 12{w rSub { size 8{"fl"} } } {} is the weight of the fluid displaced by the object.

Humm … High-tech body swimsuits were introduced in 2008 in preparation for the Beijing Olympics. One concern (and international rule) was that these suits should not provide any buoyancy advantage. How do you think that this rule could be verified?

Making connections: take-home investigation

The density of aluminum foil is 2.7 times the density of water. Take a piece of foil, roll it up into a ball and drop it into water. Does it sink? Why or why not? Can you make it sink?

Floating and sinking

Drop a lump of clay in water. It will sink. Then mold the lump of clay into the shape of a boat, and it will float. Because of its shape, the boat displaces more water than the lump and experiences a greater buoyant force. The same is true of steel ships.

Calculating buoyant force: dependency on shape

(a) Calculate the buoyant force on 10,000 metric tons ( 1 . 00 × 10 7 kg ) size 12{ \( 1 "." "00" times "10" rSup { size 8{7} } `"kg" \) } {} of solid steel completely submerged in water, and compare this with the steel’s weight. (b) What is the maximum buoyant force that water could exert on this same steel if it were shaped into a boat that could displace 1 . 00 × 10 5 m 3 size 12{1 "." "00" times "10" rSup { size 8{5} } `m rSup { size 8{3} } } {} of water?

Strategy for (a)

To find the buoyant force, we must find the weight of water displaced. We can do this by using the densities of water and steel given earlier (see "Density"). We note that, since the steel is completely submerged, its volume and the water’s volume are the same. Once we know the volume of water, we can find its mass and weight.

Solution for (a)

First, we use the definition of density ρ = m V size 12{ρ= { {m} over {V} } } {} to find the steel’s volume, and then we substitute values for mass and density. This gives

V st = m st ρ st = 1 . 00 × 10 7 kg 7 . 8 × 10 3 kg/m 3 = 1 . 28 × 10 3 m 3 . size 12{v rSub { size 8{"st"} } = { {m rSub { size 8{"st"} } } over {ρ rSub { size 8{"st"} } } } = { {1 "." "00" times "10" rSup { size 8{7} } `"kg"} over {7 "." 8 times "10" rSup { size 8{3} } `"kg/m" rSup { size 8{3} } } } =1 "." "28" times "10" rSup { size 8{3} } `m rSup { size 8{3} } } {}

Because the steel is completely submerged, this is also the volume of water displaced, V w size 12{V rSub { size 8{w} } } {} . We can now find the mass of water displaced from the relationship between its volume and density, both of which are known. This gives

m w = ρ w V w = ( 1.000 × 10 3 kg/m 3 ) ( 1.28 × 10 3 m 3 ) = 1.28 × 10 6 kg. alignc { stack { size 12{m rSub { size 8{w} } =ρ rSub { size 8{w} } V rSub { size 8{w} } = \( 1 "." "000" times "10" rSup { size 8{3} } `"kg/m" rSup { size 8{3} } \) \( 1 "." "28" times "10" rSup { size 8{3} } `m rSup { size 8{3} } \) } {} #=1 "." "28" times "10" rSup { size 8{6} } `"kg" "." {} } } {}

By Archimedes’ principle, the weight of water displaced is m w g size 12{m rSub { size 8{w} } g} {} , so the buoyant force is

F B = w w = m w g = 1.28 × 10 6 kg 9.80 m/s 2 = 1.3 × 10 7 N. alignc { stack { size 12{F rSub { size 8{B} } =w rSub { size 8{w} } =m rSub { size 8{w} } g= left (1 "." "28" times "10" rSup { size 8{6} } `"kg" right ) left (9 "." "80"`"m/s" rSup { size 8{2} } right )} {} #=1 "." 3 times "10" rSup { size 8{7} } `N "." {} } } {}

The steel’s weight is m w g = 9 . 80 × 10 7 N size 12{m rSub { size 8{w} } g=9 "." "80" times "10" rSup { size 8{7} } `N} {} , which is much greater than the buoyant force, so the steel will remain submerged. Note that the buoyant force is rounded to two digits because the density of steel is given to only two digits.

Strategy for (b)

Here we are given the maximum volume of water the steel boat can displace. The buoyant force is the weight of this volume of water.

Solution for (b)

The mass of water displaced is found from its relationship to density and volume, both of which are known. That is,

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Concepts of physics. OpenStax CNX. Aug 25, 2015 Download for free at https://legacy.cnx.org/content/col11738/1.5
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