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Examples of harmonic motion are :

  • x = A e - ω t sin ω t
  • x = A sin ω t
  • x = A ω cos ω t
  • x = A sin ω t + B cos ω t

Of these, last three examples are simple harmonic oscillations.

Note that we can reduce fourth example, sum of two trigonometric functions, into a single trigonometric function with appropriate substitutions. As a matter of fact, we shall illustrate such reduction in appropriate context.

The simple harmonic oscillation is popularly known as simple harmonic motion (SHM). The important things to reemphasize here is that SHM denotes an oscillation, which does not involve change in amplitude. We shall learn that this represents a system in which energy is not dissipated. It means that mechanical energy of a system in SHM is conserved.

Non-harmonic oscillation

A non-harmonic oscillation is one, which is not harmonic motion. We can consider combination of two or more harmonic motions of different frequencies as an illustration of non-harmonic function.

x = A sin ω t + B sin 2 ω t

We can not reduce this sum into a single trigonometric sine or cosine function and as such, motion described by the function is non-harmonic.

Simple harmonic motion

A simple harmonic motion can be conceived as a “to and fro” motion along an axis (say x-axis). In order to simplify the matter, we choose origin of the reference as the point about which particle oscillates. If we start our observation from positive extreme of the motion, then displacement of the particle “x” at a time “t” is given by :

x = A cos ω t

where “ω” is angular frequency and “t” is the time. The figure here shows the positions of the particle executing SHM at an interval of “T/8”. The important thing to note here is that displacements in different intervals are not equal, suggesting that velocity of the particle is not uniform. This also follows from the nature of cosine function. The values of cosine function are not equally spaced with respect to angles.

Simple harmonic motion

Positions of the particles at different times are shown.


We know that value of cosine function lies between “-1” and “1”. Hence, value of “x” varies between “-A” and “A”. If we plot the function describing displacement, then the plot is similar to that of cosine function except that its range of values lies between “-A” and “A”.


The scalar value of maximum displacement from the mean position is known as the amplitude of oscillation.

The value “A” denotes the maximum displacement in either direction. The scalar value of maximum displacement from the mean position is known as the amplitude of oscillation. If we consider pendulum, we can observe that farther is the point from which pendulum bob (within the permissible limit in which the bob executes SHM) is released, greater is the amplitude of oscillation. Similarly, greater is the stretch or compression in the spring executing SHM, greater is the amplitude. Alternatively, we can say that greater is the force causing motion, greater is the amplitude. In the nutshell, amplitude of SHM depends on the initial conditions of motion - force and displacement.

Questions & Answers

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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
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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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