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x = A 1 + A 2 cos φ sin ω t + A 2 sin φ cos ω t

The expressions in the brackets are constant. Let,

C = A 1 + A 2 cos φ and D = A 2 sin φ

Substituting in the expression of displacement, we have :

x = C sin ω t + D cos ω t

Following standard analytical method, Let

C = A cos θ and D = A sin θ

Substituting in the expression of displacement again, we have :

x = A cos θ sin ω t + A sin θ cos ω t

x = A sin ω t + θ

This is the final expression of the composition of two SHMs in the same straight line. Clearly, the amplitude of resulting SHM is “A”. Also, the resulting SHM differs in phase with respect to either of the two SHMs. In particular, the phase of resulting SHM differs by an angle “θ” with respect of first SHM, whose displacement is given by “ A 1 sin ω t ” We also note that frequency of the resulting SHM is same as either of two SHMs.

Phase constant

The phase constant of the resulting SHM is :

tan θ = D C = A 2 sin φ A 1 + A 2 cos φ


The amplitude of the resultant harmonic motion is obtained solving substitutions made in the derivation.

C = A cos θ and D = A sin θ

A = C 2 + D 2 = { A 1 + A 2 cos φ 2 + A 2 sin φ 2 }

A = A 1,2 + A 2 2 cos 2 φ + 2 A 1 A 2 cos φ + A 2 2 sin 2 φ

A = A 1 2 + A 2 2 + 2 A 1 A 2 cos φ

Important cases

We ,here, consider few interesting cases :

1: phase difference is zero

Two SHMs are in same phase. In this case, cos φ = cos 0 0 = 1 .

A = A 1 2 + A 2 2 + 2 A 1 A 2 cos φ = A 1 2 + A 2 2 + 2 A 1 A 2 = A 1 + A 2 2

A = A 1 + A 2

If additionally A 1 = A 2 , then A = 2 A 1 = 2 A 2 . Further, phase constant is given by :

tan θ = D C = A 2 sin φ A 1 + A 2 cos φ = A 2 sin 0 0 A 1 + A 2 cos 0 0 = 0

θ = 0

2: phase difference is “π”

Two SHMs are opposite in phase. In this case, cos φ = cos π = - 1 .

A = A 1 2 + A 2 2 + 2 A 1 A 2 cos φ = A 1 2 + A 2 2 2 A 1 A 2 = A 1 A 2 2

A = A 1 A 2

The amplitude is a non-negative number. In order to reflect this aspect, we write amplitude in modulus form :

A = | A 1 A 2 |

If additionally A 1 = A 2 , then A = 0. In this case, the particle does not oscillate. Further, phase constant is given by :

tan θ = D C = A 2 sin φ A 1 + A 2 cos φ = A 2 sin π A 1 + A 2 cos π = 0

θ = 0

Composition by vector method

We have evaluated composition of two SHMs analytically. This has given us the detailed picture of how displacement of a particle takes place. In the nutshell, we find that resulting motion is also a SHM of same frequency as that of constituting SHMs. Besides, we are able to determine followings aspects of resulting SHM :

  • Displacement
  • Amplitude of the resulting SHM
  • Phase constant of the resulting SHM

There is, however, an effective and more convenient alternative to determine all these aspects of SHM, using vector concept. The important thing to realize here is that amplitude can be associated with direction – apart from having magnitude. Its direction is qualified by the phase constant.

The equation of amplitude, derived earlier, provides the basis of this assumption. If we look closely at the expression of resultant amplitude, then we realize that the expression actually represents sum of two vectors namely “ A 1 ” and “ A 2 ” at an angle “φ” as shown in the figure.

Composition of two shms

The diagonal represents the resultant amplitude.

A = A 1 2 + A 2 2 + 2 A 1 A 2 cos φ

This understanding serves our purpose. If we know amplitudes of individual SHMs and the phase difference, then we can find amplitude of the resulting SHM directly using vector sum formula. It is also evident that vector method can be used to find the resulting phase difference. From the figure,

Questions & Answers

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?
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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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.
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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?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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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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