# 10.4 Energy and the simple harmonic oscillator

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• Determine the maximum speed of an oscillating system.

To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have We know from Hooke’s Law: Stress and Strain Revisited that the energy stored in the deformation of a simple harmonic oscillator is a form of potential energy given by:

${\text{PE}}_{\text{el}}=\frac{1}{2}{\mathit{kx}}^{2}.$

Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy $\text{KE}$ . Conservation of energy for these two forms is:

$\text{KE}+{\text{PE}}_{\text{el}}=\text{constant}$

or

$\frac{1}{2}{\text{mv}}^{2}+\frac{1}{2}{\text{kx}}^{2}=\text{constant.}$

This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role

Namely, for a simple pendulum we replace the velocity with $v=\mathrm{L\omega }$ , the spring constant with $k=\text{mg}/L$ , and the displacement term with $x=\mathrm{L\theta }$ . Thus

$\frac{1}{2}{\text{mL}}^{2}{\omega }^{2}+\frac{1}{2}\text{mgL}{\theta }^{2}=\text{constant.}$

In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates. So for the simple example of an object on a frictionless surface attached to a spring, as shown again in [link] , the motion starts with all of the energy stored in the spring. As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.

The conservation of energy principle can be used to derive an expression for velocity $v$ . If we start our simple harmonic motion with zero velocity and maximum displacement ( $x=X$ ), then the total energy is

$\frac{1}{2}{\text{kX}}^{2}.$

This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each. The conservation of energy for this system in equation form is thus:

$\frac{1}{2}{\text{mv}}^{2}+\frac{1}{2}{\text{kx}}^{2}=\frac{1}{2}{\text{kX}}^{2}.$

Solving this equation for $v$ yields:

$v=±\sqrt{\frac{k}{m}\left({X}^{2}-{x}^{2}\right)}.$

Manipulating this expression algebraically gives:

$v=±\sqrt{\frac{k}{m}}X\sqrt{1-\frac{{x}^{2}}{{X}^{2}}}$

and so

$v=±{v}_{\text{max}}\sqrt{1-\frac{{x}^{2}}{{X}^{2}}},$

where

${v}_{\text{max}}=\sqrt{\frac{k}{m}}X.$

From this expression, we see that the velocity is a maximum ( ${v}_{\text{max}}$ ) at $x=0$ , as stated earlier in $v\left(t\right)=-{v}_{\text{max}}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\frac{2\pi t}{T}$ . Notice that the maximum velocity depends on three factors. Maximum velocity is directly proportional to amplitude. As you might guess, the greater the maximum displacement the greater the maximum velocity. Maximum velocity is also greater for stiffer systems, because they exert greater force for the same displacement. This observation is seen in the expression for ${v}_{\text{max}};$ it is proportional to the square root of the force constant $k$ . Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of $m$ . For a given force, objects that have large masses accelerate more slowly.

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
learn
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
what is variations in raman spectra for nanomaterials
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
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
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Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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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 ?
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?
?
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
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