Composition Of Two SHM - Practice Questions & MCQ

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_1 \sin \omega t
$

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

$
x_2=A \sin (\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+2 A_1 A_2 \cdot \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^{\prime}=\tan ^{-1}\left(\frac{A_2 \sin \phi}{A_1+A_2 \cos \phi}\right)
$
 

Composition of SHM in perpendicular direction:

Let a force $\mathrm{F}_1$ on a particle produces an SHM given by

$
x=A \sin \omega t
$

and a force $\mathrm{F}_2$ alone produces an SHM given by

$
x=A \sin (\omega t+\phi)
$

- Both the force $F_1$ and $F_2$ acting 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{2 x y \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} \cdot 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} \cdot 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