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How to calculate the energy in a simple harmonic oscillator.

Energy in sho

Recall that the total energy of a system is: E = K E + P E = K + U We also know that the kinetic energy is

K = 1 2 m v 2 But what is U ? For a conservative Force( F d x = 0 ) - eg. gravity, electrical... (no friction) we know that the work done by anexternal force is stored as U . For the case of a mass on a spring, the external force is opposite the springForce (That is it has the opposite sign from the spring force).: F e x t = k x (i.e. This is the force you use to pull the mass and stretch the spring beforeletting go and making it oscillate.)

Thus U = 0 x k x x = 1 2 k x 2 This gives: E = 1 2 m v 2 + 1 2 k x 2 = 1 2 m ( x t ) 2 + 1 2 k x 2 It is important to realize that any system that is represented by either of these two equations below represents oscillating system m 2 x t 2 + k x = 0 1 2 m ( x t ) 2 + 1 2 k x 2 = E

To calculate the energy in the system it is helpful to take advantage of the fact that we can calculate the energy at any point in x. For example in thecase of the simple harmonic oscillator we have that: x = A e i ( ω t + α ) We can choose t such that x = A Now remember that when I write x = A e i ( ω t + α ) I "really" (pun intended) mean x = R e [ A e i ( ω t + α ) ] Likewise then x ˙ = R e [ i ω A e i ( ω t + α ) ]

At the point in time where x = A this gives us x ˙ = R e [ i ω A ] = 0 Thus at that point in time we have x ˙ = 0 . We can now substitute that and x = A into E = 1 2 m ( x t ) 2 + 1 2 k x 2 we obtain E = 1 2 k A 2 This is an important point. The energy in the oscillator is proportional to the amplitude squared!

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
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