Step 1: Understanding the Question:
The given equation represents the superposition of two perpendicular SHMs of the same frequency, or more simply, a combination of two sine and cosine functions. Step 2: Key Formula or Approach:
For $x = a_1 \sin \omega t + a_2 \cos \omega t$, the resultant amplitude is $R = \sqrt{a_1^2 + a_2^2}$. Step 3: Detailed Explanation:
Let $A = R \cos \theta$ and $B = R \sin \theta$.
The equation becomes:
\[ x = R \cos \theta \sin \omega t + R \sin \theta \cos \omega t \]
\[ x = R \sin(\omega t + \theta) \]
This is the standard equation of SHM with amplitude $R$.
From the substitution:
\[ A^2 + B^2 = R^2 \cos^2 \theta + R^2 \sin^2 \theta = R^2(\cos^2 \theta + \sin^2 \theta) = R^2 \]
\[ R = \sqrt{A^2 + B^2} = (A^2 + B^2)^{1/2} \] Step 4: Final Answer:
The motion is simple harmonic with amplitude $(A^2 + B^2)^{1/2}$.
