Question:
A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; \(x = a_1\cos\omega t\) and \(y = a_2\cos 2\omega t\) traces a curve given by :
A particle which is simultaneously subjected to two perpendicular simple harmonic motions represented by; \(x = a_1\cos\omega t\) and \(y = a_2\cos 2\omega t\) traces a curve given by :
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Eliminate \(t\) with \(\cos2\omega t=2\cos^2\omega t-1\) to get \(y=a_2\left(\dfrac{2x^2}{a_1^2}-1\right)\), a parabola.
Updated On: Nov 11, 2026
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The Correct Option is D
Solution and Explanation
\(y=a_2\cos2\omega t=a_2(2\cos^2\omega t-1)=a_2\left(\dfrac{2x^2}{a_1^2}-1\right)\).
This is a parabola opening upwards (along \(+y\)) with vertex at \(x=0,\ y=-a_2\), reaching \(y=+a_2\) at \(x=\pm a_1\) and crossing \(y=0\) at \(x=\pm a_1/\sqrt2\).
This is the U-shaped curve with its lowest point on the \(y\)-axis at \(-a_2\), shown in option (4).
Answer: (4) the upward-opening parabola with vertex at \((0,-a_2)\)
This is a parabola opening upwards (along \(+y\)) with vertex at \(x=0,\ y=-a_2\), reaching \(y=+a_2\) at \(x=\pm a_1\) and crossing \(y=0\) at \(x=\pm a_1/\sqrt2\).
This is the U-shaped curve with its lowest point on the \(y\)-axis at \(-a_2\), shown in option (4).
Answer: (4) the upward-opening parabola with vertex at \((0,-a_2)\)
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