What is the proof that the superposition of two mutually perpendicular simple harmonic motions (SHMs) with the same angular frequency but different magnitudes and initial phases can produce elliptic motion?
a. Use the equation for elliptic motion, x = A×cos(ωt + φ) and y = B×sin(ωt + φ), where A and B are the amplitudes of the two SHMs, ω is the angular frequency, t is time, and φ is the phase angle.

b. Substituting the equations for x and y into the equation for elliptic motion, we get x²/A² + y²/B² = 1, which represents an ellipse.

c. Use the fact that the sum of two perpendicular SHMs is elliptic motion. The superposition of two perpendicular SHMs with the same angular frequency but different magnitudes and initial phases leads to elliptic motion.

d. Mathematically, the superposition of the two SHMs can be represented as a vector sum, and the resulting motion can be shown to be elliptic in nature.