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Question: When two displacements represented by \( Y_1 = a\ sin(\omega t) \ and\ Y_2 = bcos(\omega t) \) are s...

When two displacements represented by Y1=a sin(ωt) and Y2=bcos(ωt)Y_1 = a\ sin(\omega t) \ and\ Y_2 = bcos(\omega t) are superimposed, the motion is:
A. Simple harmonic with amplitude a2+b2\sqrt{a^2+b^2}
B. Simple harmonic with amplitude (a+b2)(\dfrac{a+b}{2})
C. Not a simple harmonic
D. Simple harmonic with amplitude ab\dfrac ab.

Explanation

Solution

A motion in which a particle undergoes periodic motion is called Simple harmonic motion (S.H.M). Not every periodic motion is S.H.M but every S.H.M is periodic motion. The revolution of earth about the sun is an example of periodic motion but it is not simple harmonic. A motion is said to be simple harmonic only if the acceleration of the particle is the function of first power of displacement and having direction opposite of the displacement.

Complete step-by-step answer:
First, let’s understand the standard S.H.M equation.
Y=asin(ωt+ϕ)Y=asin\left( \omega t+\phi \right) is called the standard S.H.M equation.
Here ‘Y’ represents the displacement of wave particles at time ‘t’. Coefficient of trigonometric function ‘a’ is called the amplitude of the wave.’ ω\omega ’ is the angular frequency of the wave, which is the measure of angular displacement. ‘ ϕ\phi ’ is the initial phase difference of the wave. It is also called ‘epoch’.
Mathematically we can say that if the motion is simple harmonic, it must follow the standard differential equation of simple harmonic motion which is given by d2xdt2=−ω2x\dfrac{d^2x}{dt^2} = -\omega^2 x.
The solutions of this equation comes in the form of sin and cos.
Hence clearly the equations represent simple harmonic motion.
Now, after superimposition of these shm, we get the resultant shm as:
y=asinωt+bcosωty=asin\omega t + bcos \omega t
  ⟹  y=a2+b2[aa2+b2sinωt+ba2+b2cosωt]\implies y = \sqrt{a^2+b^2}\left[\dfrac{a}{\sqrt{a^2+b^2}}sin\omega t + \dfrac{b}{\sqrt{a^2+b^2}}cos\omega t\right]
  ⟹  y=a2+b2(cosϕsinωt+sinϕcosωt)=a2+b2sin(ωt+ϕ)\implies y=\sqrt{a^2+b^2} (cos\phi sin\omega t + sin \phi cos \omega t) = \sqrt{a^2+b^2}sin(\omega t + \phi) [where ϕ=tan−1[ba]\phi = tan^{-1} \left[\dfrac ba\right] ]
Which is also the equation of SHM having amplitude a2+b2\sqrt{a^2+b^2}

So, the correct answer is “Option A”.

Note: This superposition of two or more simple harmonic motions is also called interference of waves. Since every particle constituting the wave also executes simple harmonic motion, thus a wave is the best example of SHM. The superposition of waves is their very important property.