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.

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.

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

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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