To determine the nature of the motion when two displacements are superimposed, we need to analyze the given expressions:$\\ y_1 = a\sin(\omega t) \\ y_2 = b\cos(\omega t) \\$The resultant displacement $y$ is given by the superposition of $y_1$ and $y_2: \\ y = y_1 + y_2 = a\sin(\omega t) + b\cos(\omega t) \\$This expression can be rewritten in the form of a single sinusoidal function using the identity:$\\ R\sin(\omega t + \phi) = R\left(\sin(\omega t)\cos(\phi) + \cos(\omega t)\sin(\phi)\right) \\$Comparing coefficients, we have:$\\ a = R\cos(\phi) \\ b = R\sin(\phi) \\$To find $R,$ we use the Pythagorean identity:$\\ R^2 = a^2 + b^2 \\ R = \sqrt{a^2 + b^2} \\$Thus, the resultant motion is simple harmonic with amplitude $\sqrt{a^2 + b^2}. \\$Therefore, the correct option is Option 2.