# 8.6 Archimedes’ principle  (Page 2/9)

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${F}_{\text{B}}={w}_{\text{fl}},$

where ${F}_{\text{B}}$ is the buoyant force and ${w}_{\text{fl}}$ 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

${F}_{\text{B}}={w}_{\text{fl}},$

where ${F}_{\text{B}}$ is the buoyant force and ${w}_{\text{fl}}$ 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 $\left(1\text{.}\text{00}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)$ 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 $1\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}$ 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 in [link] . 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 $\rho =\frac{m}{V}$ to find the steel’s volume, and then we substitute values for mass and density. This gives

${V}_{\text{st}}=\frac{{m}_{\text{st}}}{{\rho }_{\text{st}}}=\frac{1\text{.}\text{00}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{kg}}{7\text{.}8×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}}=1\text{.}\text{28}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}.$

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

$\begin{array}{lll}{m}_{w}& =& {\rho }_{w}{V}_{w}=\left(\text{1.000}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left(1.28×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{m}^{3}\right)\\ & =& \text{1.28}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{kg.}\end{array}$

By Archimedes’ principle, the weight of water displaced is ${m}_{\text{w}}g$ , so the buoyant force is

$\begin{array}{lll}{F}_{B}& =& {w}_{w}={m}_{w}g=\left(\text{1.28}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)\\ & =& 1.3×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N.}\end{array}$

The steel’s weight is ${m}_{\text{w}}g=9\text{.}\text{80}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{N}$ , 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,

$\begin{array}{lll}{m}_{w}& =& {\rho }_{w}{V}_{w}=\left(\text{1.000}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left(\text{1.00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}{m}^{3}\right)\\ & =& \text{1.00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{kg.}\end{array}$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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