When two displacements represented by $y_1 = a \sin (\omega t)$ and $y_2 = b \cos (\omega t)$ are…
When two displacements represented by $y_1 = a \sin (\omega t)$ and $y_2 = b \cos (\omega t)$ are superimposed, the motion is [CBSE AIPMT 2015]
not a simple harmonic
simple harmonic with amplitude $\frac{a}{b}$
simple harmonic with amplitude $\sqrt{a^2 + b^2}$
simple harmonic with amplitude $\frac{(a + b)}{2}$
Solution
Given, $y_1 = a \sin \omega t$ and $y_2 = b \cos \omega t = b \sin \left(\omega t + \frac{\pi}{2}\right)$
The resultant displacement is given by
$y = y_1 + y_2 = \sqrt{a^2 + b^2} \sin(\omega t + \phi)$
Hence, the motion of superimposed wave is simple harmonic with amplitude $\sqrt{a^2 + b^2}$.