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The subject matter of this module is linear SHM – harmonic motion along a straight line about the point of oscillation. There are various physical quantities associated with simple harmonic motion. Here, we intend to have a closer look at quantities associated with SHM like velocity, acceleration, work done, kinetic energy, potential energy and mechanical energy etc. For the sake of completeness, we shall also have a recap of concepts already discussed in earlier modules.

The SHM force relation “F = -kx” is a generic form of equation for linear SHM – not specific to block-spring system. In the case of block-spring system, “k” is the spring constant. This point is clarified to emphasize that relations that we shall be developing in this module applies to all linear SHM and not to a specific case.

Since displacement of SHM can be represented either in cosine or sine forms, depending where we start observing motion at t = 0. For someone, it is easier to visualize beginning of SHM, when particle is released from positive extreme. On the other hand, expression in sine form is convenient as particle is at the center of oscillation at t = 0. For this reason, some prefer sine representation.

The very fact that there are two ways to represent displacement may pose certain ambiguity or uncertainty in mind. We shall , therefore, strive to maintain complete independence of forms with the understanding that when it is cosine function, then starting reference is positive extreme and if it is sine function, then starting reference is center of oscillation. In order to illustrate flexibility, we shall be using “sine” expression of displacement in this module instead of cosine function, which has so far been used.

Displacement

The displacement of the particle is given by :

x = A sin ω t + φ

where “A” is the amplitude,"ω" is angular frequency, “φ” is the phase constant and “ωt + φ” is the phase. Clearly, displacement is periodic with respect to time as it is represented by bounded trigonometric function. The displacement “x” varies between “-A” and “A”.

Velocity

The velocity of the particle as obtained from the solution of SHM equation is given by :

v = ω A 2 x 2

This is the relation of velocity of the particle with respect to displacement along the path of oscillation, bounded between “-ωA” and “ωA”. We can obtain a relation of velocity with respect to time by substituting expression of displacement “x” in the above equation :

v = ω A 2 x 2 = ω { A 2 A 2 sin 2 ω t + φ } = ω A cos ω t + φ

We can ,alternatively, deduce this expression by differentiating displacement, “x”, with respect to time :

v = x t = t A sin ω t + φ = ω A cos ω t + φ

The variation of velocity with respect to time is sinusoidal and hence periodic. Here, we draw both displacement and velocity plots with respect to time in order to compare how velocity varies as particle is at different positions.

Velocity - time plot

The velocity is represented by cosine function.

The upper figure is displacement – time plot, whereas lower figure is velocity – time plot. We observe following important points about variation of velocity :

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
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Damian Reply
absolutely yes
Daniel
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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
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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
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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.
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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
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
Porter
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Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
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