# 7.3 Applications  (Page 4/13)

 Page 4 / 13

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.

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

Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
Got questions? Join the online conversation and get instant answers!