# 10.3 The physics of springs  (Page 3/11)

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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:

${A}^{T}KAx=f.$

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

$KAx=Ke=\left[\begin{array}{c}{k}_{1}\\ & \ddots \\ & & {k}_{16}\end{array}\right]\left[\begin{array}{c}{e}_{1}\\ ⋮\\ {e}_{16}\end{array}\right]=\left[\begin{array}{c}{e}_{1}\\ & \ddots \\ & & {e}_{16}\end{array}\right]\left[\begin{array}{c}{k}_{1}\\ ⋮\\ {k}_{16}\end{array}\right].$

This simple substitution lets us rewrite our equation as

${A}^{T}diag\left({e}^{\left(i\right)}\right)k={f}^{\left(i\right)}-{f}^{\left(0\right)}$

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

$B=\left[\begin{array}{c}{A}^{T}diag\left({e}^{\left(1\right)}\right)\\ {A}^{T}diag\left({e}^{\left(2\right)}\right)\\ ⋮\\ {A}^{T}diag\left({e}^{\left(s\right)}\right)\end{array}\right],\phantom{\rule{1.em}{0ex}}f=\left[\begin{array}{c}{f}^{\left(1\right)}-{f}^{\left(0\right)}\\ {f}^{\left(2\right)}-{f}^{\left(0\right)}\\ ⋮\\ {f}^{\left(s\right)}-{f}^{\left(0\right)}\end{array}\right].$

We now recall Hooke's Law:

$f=Bk.$

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

$\underset{k\in {\mathbb{R}}^{16}}{min}{\parallel Bk-f\parallel }^{2}.$

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

${B}^{T}Bk={B}^{T}f.$

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

$k={\left({B}^{T}B\right)}^{-1}{B}^{T}f.$

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.

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, $\pi /4$ , or $\pi /2$ 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.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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