<< Chapter < Page Chapter >> Page >

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 ( t ) = 0.6 e −2 t 0.2 e −6 t

Got questions? Get instant answers now!

Case 2: b 2 = 4 m k

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

x ( t ) = c 1 e λ 1 t + c 2 t e λ 1 t ,

where λ 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 4 m k are exactly equal. [link] shows what typical critically damped behavior looks like.

This figure has two graphs labeled (a) and (b). The first graph is in the first quadrant and is a decreasing curve with the horizontal axis as a horizontal asymptote. The second graph initially is a decreasing function but becomes increasing below the horizontal axis. Then, the horizontal axis is also a horizontal asymptote.
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 m g = 1 ( 9.8 ) = 0.2 k , so k = 49 . Then, the differential equation is

x + 14 x + 49 x = 0 ,

which has general solution

x ( t ) = c 1 e −7 t + c 2 t e −7 t .

Applying the initial conditions x ( 0 ) = 0 and x ( 0 ) = −3 gives

x ( t ) = −3 t e −7 t .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 ( t ) = 1 2 e −8 t + 4 t e −8 t

Got questions? Get instant answers now!

Case 3: b 2 < 4 m k

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

x ( t ) = e α t ( c 1 cos ( β t ) + c 2 sin ( β t ) ) ,

where α 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.

This figure is an oscillating graph where the amplitude is decreasing. There are red dashed curves at the peaks of the amplitudes showing the pattern of a decreasing amplitude. As t increases, the horizontal axis becomes a horizontal asymptote.
Behavior of an underdamped spring-mass system.

Note that for all damped systems, lim t x ( t ) = 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 = 16 3.2 = 5 and m = 16 32 = 1 2 , so the differential equation is

1 2 x + x + 5 x = 0 , or x + 2 x + 10 x = 0 .

This equation has the general solution

x ( t ) = e t ( c 1 cos ( 3 t ) + c 2 sin ( 3 t ) ) .

Applying the initial conditions, x ( 0 ) = 3 4 and x ( 0 ) = 0 , we get

x ( t ) = e t ( 3 4 cos ( 3 t ) + 1 4 sin ( 3 t ) ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 ( t ) = −0.24 e −2 t cos ( 4 t ) 0.12 e −2 t sin ( 4 t )

Got questions? Get instant answers now!

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
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask