# 7.3 Applications  (Page 4/13)

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A 2-kg mass is attached to a spring with spring constant 24 N/m. The system is then immersed in a medium imparting a damping force equal to 16 times the instantaneous velocity of the mass. Find the equation of motion if it is released from rest at a point 40 cm below equilibrium.

$x\left(t\right)=0.6{e}^{-2t}-0.2{e}^{-6t}$

## Case 2: ${b}^{2}=4mk$

In this case, we say the system is critically damped . The general solution has the form

$x\left(t\right)={c}_{1}{e}^{{\lambda }_{1}t}+{c}_{2}t{e}^{{\lambda }_{1}t},$

where ${\lambda }_{1}$ is less than zero. The motion of a critically damped system is very similar to that of an overdamped system. It does not oscillate. However, with a critically damped system, if the damping is reduced even a little, oscillatory behavior results. From a practical perspective, physical systems are almost always either overdamped or underdamped (case 3, which we consider next). It is impossible to fine-tune the characteristics of a physical system so that ${b}^{2}$ and $4mk$ are exactly equal. [link] shows what typical critically damped behavior looks like. Behavior of a critically damped spring-mass system. The system graphed in part (a) has more damping than the system graphed in part (b).

## Critically damped spring-mass system

A 1-kg mass stretches a spring 20 cm. The system is attached to a dashpot that imparts a damping force equal to 14 times the instantaneous velocity of the mass. Find the equation of motion if the mass is released from equilibrium with an upward velocity of 3 m/sec.

We have $mg=1\left(9.8\right)=0.2k,$ so $k=49.$ Then, the differential equation is

$x\text{″}+14{x}^{\prime }+49x=0,$

which has general solution

$x\left(t\right)={c}_{1}{e}^{-7t}+{c}_{2}t{e}^{-7t}.$

Applying the initial conditions $x\left(0\right)=0$ and ${x}^{\prime }\left(0\right)=-3$ gives

$x\left(t\right)=-3t{e}^{-7t}.$

A 1-lb weight stretches a spring 6 in., and the system is attached to a dashpot that imparts a damping force equal to half the instantaneous velocity of the mass. Find the equation of motion if the mass is released from rest at a point 6 in. below equilibrium.

$x\left(t\right)=\frac{1}{2}{e}^{-8t}+4t{e}^{-8t}$

## Case 3: ${b}^{2}<4mk$

In this case, we say the system is underdamped . The general solution has the form

$x\left(t\right)={e}^{\alpha t}\left({c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(\beta t\right)+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(\beta t\right)\right),$

where $\alpha$ is less than zero. Underdamped systems do oscillate because of the sine and cosine terms in the solution. However, the exponential term dominates eventually, so the amplitude of the oscillations decreases over time. [link] shows what typical underdamped behavior looks like.

Note that for all damped systems, $\underset{t\to \infty }{\text{lim}}x\left(t\right)=0.$ The system always approaches the equilibrium position over time.

## Underdamped spring-mass system

A 16-lb weight stretches a spring 3.2 ft. Assume the damping force on the system is equal to the instantaneous velocity of the mass. Find the equation of motion if the mass is released from rest at a point 9 in. below equilibrium.

We have $k=\frac{16}{3.2}=5$ and $m=\frac{16}{32}=\frac{1}{2},$ so the differential equation is

$\frac{1}{2}x\text{″}+{x}^{\prime }+5x=0,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x\text{″}+2{x}^{\prime }+10x=0.$

This equation has the general solution

$x\left(t\right)={e}^{\text{−}t}\left({c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(3t\right)+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(3t\right)\right).$

Applying the initial conditions, $x\left(0\right)=\frac{3}{4}$ and ${x}^{\prime }\left(0\right)=0,$ we get

$x\left(t\right)={e}^{\text{−}t}\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(3t\right)+\frac{1}{4}\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(3t\right)\right).$

A 1-kg mass stretches a spring 49 cm. The system is immersed in a medium that imparts a damping force equal to four times the instantaneous velocity of the mass. Find the equation of motion if the mass is released from rest at a point 24 cm above equilibrium.

$x\left(t\right)=-0.24{e}^{-2t}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(4t\right)-0.12{e}^{-2t}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(4t\right)$

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
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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
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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
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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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?
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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?
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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 .
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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?
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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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