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Our solution to this problem involves experimental stacking. Due to the fact that our experiment is easily repeatable under different conditions, we can arrive at more observations by simply altering the forces acting on the table. This, however, should not alter any of the spring constants. By doing so, we generate a second set of data which is still dependent on the spring constants. We are left with 16 springs, but we now have 28 degrees of freedom: an overdetermined system. The experiment can be repeated s times to minimize experimental error as well, yielding 14 s degrees of freedom and the same 16 values in K .

With this experimental technique in hand, we can revisit our primary equation:

A T K A x = f .

In order to solve for K , we re-express part of our equation:

K A x = K e = k 1 k 16 e 1 e 16 = e 1 e 16 k 1 k 16 .

This simple substitution lets us rewrite our equation as

A T diag e ( i ) k = f ( i ) - f ( 0 )

which allows us to solve for k .

At this point, we can apply our experimental technique of stacking. Using s experiments, we may construct the appropriate matrices such that

B = A T diag e ( 1 ) A T diag e ( 2 ) A T diag e ( s ) , f = f ( 1 ) - f ( 0 ) f ( 2 ) - f ( 0 ) f ( s ) - f ( 0 ) .

We now recall Hooke's Law:

f = B k .

With f and B , we are now ready to solve for k . However, due to the fact that our system is overdetermined, there does not have to be a unique solution. To find the solution of best fit, then, we turn to the least squares method so as to find k that satisfies

min k R 16 B k - f 2 .

We go about this using the standard method of normal equations: we multiply both sides of the Hooke equation by B T :

B T B k = B T f .

This allows us to use the Moore-Penrose psuedoinverse of B solve for k , our vector of spring constants:

k = ( B T B ) - 1 B T f .

Bar graphs of k are presented from a laboratory implementation of this technique. [link] shows the calculated spring constants using the first two experiments from Data Set A. These spring constants have 246.7% error (see "Notes: Our Data Sets, Measuring Spring Constants, and Error" for notes on data sets and calculating error). [link] shows the measured spring constants.

Results from Stacking Two Experiments

Due to our technique stacking of s experiments, our solution should minimize experimental error. However, it should be clear that, due to the amount of experimental error involved, we are extremely unlikely to arrive at an exact solution to the problem. It is important to note that, though wildly inaccurate, we can correctly identify the stiff spring in the system.

Our question

In theory, this works out beautifully. Unfortunately, our experiments are carried out in the real world, so the entire process is rife with error. Measurements of both the masses and the positions of the nodes may be imprecise, and the alignment of the webcam may be off (See [link] ), introducing the keystone effect, all of which dirties the displacement data. The forces may not be perfectly aligned along the horizontal and vertical (See [link] ), and the masses we hang may not be exactly what we believe, introducing error in the force vector. Furthermore, Hooke's Law is an approximation, valid only in a certain range (although we keep our forces within that range). We also assume that the springs lie at angles of 0, π / 4 , or π / 2 to the nodes, a belief that is reflected in the adjacency matrix and not at all accurate (See [link] ). Our model also approximates the elongations of the springs linearly and assumes that the pennies are massless points, which introduces further error.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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