Composition Of Two SHM MCQ - Practice Questions with Answers

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 $S\mathrm{HM}$ 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

$$x_2=A\sin (\omega t+\varphi )$$

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

$$A=\sqrt{A_1^2+A_2^2+2A_1{A}_2\cdot \cos \varphi }$$

$A_1$ and $A_2$ are the amplitude of two SHM's. ${\varphi }_{\text{is }}$ phase difference.
Note: Here the frequency of each SHM's are the same
And the resulting phase is given by

$${\varphi }^{\prime }={\tan }^{-1}\left(\frac{A_2\sin \varphi }{A_1+A_2\cos \varphi }\right)$$
 

Composition of SHM in perpendicular direction:

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

$$x=A\sin \omega t$$

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

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

- Both the force $F_1$ and $F_2$ acting perpendicular on the particle will produce an SHM whose resultant is given by:

$$\frac{x^2}{A_1{}^2}+\frac{y_2^2}{A_2{}^2}-\frac{2xy\cos \varphi }{A_1{A}_2}={\sin }^2\varphi$$
 

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 $\varphi =0$ the resultant equation is given by

$$y=\frac{A_2}{A_1}\cdot x$$

It is a straight line with a slope
$\frac{A_2}{A_1}$ represented by the below figure

- When $\varphi ={\pi }_{\text{resultant equation }}$

$$y=\frac{-A_2}{A_1}\cdot x$$

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

- When $\varphi =\frac{\pi }{2}$ 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 $\varphi =\frac{\pi }{2}$ then it represents a circle