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# Composition Of Two SHM - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

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

• 32 Questions around this concept.

## Solve by difficulty

The displacement equations of two interfering waves are given by $y_{1}=10 \sin \left(\omega t+\frac{\pi}{3}\right) \mathrm{cm}$, $y_{2}=5[\sin \omega t+\sqrt{3} \cos \omega t] \mathrm{cm}$ respectively. The amplitude of the resultant wave is ____$\mathrm{cm}$.

When the waves $y_{1} = A \sin \omega t$ and $y_{2} = A \cos \omega t$ are superposed, then resultant amplitude will be:

Assertion: A closed pipe and an open pipe cannot be in resonance with the same tuning fork, if the length of the closed pipe is integral multiple of the length of the open pipe.

Reason: An odd number is not divisible by even numbers.

The  x-t graph of a Particle undergoing simple harmonic motion is shown below. The acceleration of the Particle at  t=8 / 3

In a zener-regulated power supply a zener diode with $v_2=6.0 \mathrm{~V}$ is used for regulation. The load current is to be $4 \mathrm{~mA}$ and the unregulated input is $10 \mathrm{~V}$. What should be the value of series resistance R ?

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Two identical thin rings each of radius $\mathrm{R}$ are coaxially placed at a distance $\mathrm{R}$. If the rings have a uniform mass distribution and each has mass $\mathrm{m_1 \: and \: m_2}$ respectively, then the work done in moving a mass $\mathrm{m}$ from centre of one ring to that of the other is

The electric field intensity produced by the radiation coming from a $100 \mathrm{~W}$ bulb at a distance of $3 \mathrm{~m} E$. The electric field intensity produced by the radiation coming from $60 \mathrm{~W}$at same distance is $\sqrt{\frac{x}{5}}$ then value of x is

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2 moles of an ideal monatomic gas is expanded according to relation $\mathrm{PT = constant}$ from its initial state $\mathrm{ (P_0V_0)}$ to the final state, due to which its pressure becomes half of the initial pressure. The change in internal energy is

The maximum efficiency of full wave rectifier is

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The work function of silver and sodium are 4.6 and 2.3 respectively. The ratio of the slope of the stopping potential versus frequency plot for silver to that of sodium 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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