Question

2. Two collinear harmonic oscillations
8sin (100 ) 1
x = πt
and 12sin (96 ) 2
x = πt
are superposed. Calculate the (i) maximum and minimum amplitudes, and (ii) the
frequency of amplitude modulation

Answer 1

Let us consider the harmonic oscillations as x₁ and x₂.

Thus, when two harmonic oscillations x₁ and x₂ will be superimposed then the resultant will be X= x₁ + x₂

If x₁= a₁sin(ωt+Ф₁) , x₂=a₂sin(ωt+Ф₂)

= X= Asin(ωt+Ф)

Phase difference = Ф= Ф₁-Ф₂

In the above question Ф= 100t-96t= 4t 

The amplitude of resultant oscillation is maximum when Ф= 360°

4t= 360 = t=90s

The amplitude of resultant oscillation is minimum when Ф= 180°

4t= 180 = t= 45s