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Question:

A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; x = a1cosωt and y = a2cos2ωt traces a curve given by :

Solution:

Given, y = a2cos 2ωt = a2(2cos²ωt - 1)
Now, from equation along x axis, cos ωt = x/a1
Substituting this value back, y = a2(2 × (x/a1)² - 1)
Hence, this describes a parabola facing upwards.