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Composition Of Two SHM - Practice Questions & MCQ

Edited By admin | Updated on Sep 25, 2023 25:23 PM | #NEET

Quick Facts

  • Composition of two SHM- part 2 is considered one the most difficult concept.

  • 26 Questions around this concept.

Solve by difficulty

Two waves are represented by the equations y1 = a sin (\omegat + kx + 0.57) m and y2 = a cos (\omegat + kx) m, where x is in meter and t in sec. The phase diference between them is

Concepts Covered - 2

Composition of two SHM: Part 1

Composition of two SHM:

If a particle is acted upon by two forces such that each force can produce SHM, then the resultant motion of the particle is a combination of SHM. 

Composition of two SHM in the same direction

Let a  force F_1 produces an SHM of amplitude A_1 whose equation is given by 

x_1=A_1sin\omega t

Let another force F_2 produce an SHM of amplitude A_2 whose equation is given by

x_2=Asin(\omega t+\phi)

Now if force F_1 \ and \ F_2 is acted on the particle in the same direction then the resultant amplitude of the combination of SHM's is given by

A= \sqrt{A{_{1}}^{2}+A{_{2}}^{2}+2A{_{1}}A_{2}.\cos \phi }

A_{1}and A_{2}  are the amplitude of two SHM's. \phi is phase difference.

Note: Here the frequency of each SHM's are the same

And the resulting phase is given by

\phi '= \tan ^{-1}\left ( \frac{A_{2}\sin \phi }{A_{1}+A_{2}\cos \phi } \right )

 

Composition of two SHM- part 2

Composition of SHM in perpendicular direction:

Let a force F1 on a particle produces an SHM given by 

x=Asin\omega t

and a force Falone produces an SHM given by

x=Asin(\omega t+\phi)

  • Both the force Fand Facting perpendicular on the particle will produce an SHM whose resultant is given by:

 

          \frac{x^{2}}{A{_{1}}^{2}}+\frac{y{_{2}}^{2}}{A{_{2}}^{2}}-\frac{2xy\cos \phi}{A_{1}A_{2}}= \sin ^{2}\phi

The above equation is the general equation of an ellipse. That is two forces acting perpendicular on a particle execute SHM along an elliptical path.

      

  • When \phi=0 resultant equation is given by

                     y= \frac{A_{2}}{A_{1}}.x

It is a straight line with slope

\frac{A_{2}}{A_{1}}   represented by the below figure

       

  • When \phi=\pi resultant equation

         y= \frac{-A_{2}}{A_{1}}.x

which is represented by below straight line with slope \frac{-A_2}{A_1}

        

  • When \phi=\frac{\pi}{2} resultant equation

\frac{x^{2}}{A{_{1}}^{2}}+\frac{y^{2}}{A{_{2}}^{2}}= 1

It represents a normal ellipse

  •  if A_1=A_2\ and \ \phi=\frac{\pi}{2} then it represents a circle

Study it with Videos

Composition of two SHM: Part 1
Composition of two SHM- part 2

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Reference Books

Composition of two SHM: Part 1

Physics Part II Textbook for Class XI

Page No. : 33

Line : 66

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