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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.


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”.


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

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
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
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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