A particle is executing two different simple harmonic motions, mutually perpendicular, of different amplitudes and having phase difference of π 2
circular
straight line
parabolic
elliptical
Solution
Multi-choise Question
D.
elliptical
Let two perpendicular simple harmonic motion s are x = A sin ωt . . . . ( 1 ) y = B sin ωt + ϕ . . . ( 2 ) from equation ( 1 ) sin ωt = x A ⇒ sin 2 ωt + cos 2 ωt = 1 ⇒ cos 2 ωt = 1 - x 2 A 2 ⇒ cosωt = 1 - x 2 A 2 2 ⇒ y B = sin ωt cosϕ + cosωt sinϕ y B - x A cosϕ = 1 - x 2 A 2 sinϕ squaring on both side y 2 B 2 + x 2 A 2 cos 2 ϕ - 2 y B x A cosϕ = 1 - x 2 A 2 sin 2 ϕ y 2 B 2 + x 2 A 2 cos 2 ϕ + sin 2 ϕ - 2 xY AB cos ϕ = sin 2 ϕ ⇒ y 2 B 2 + x 2 A 2 - 2 xy AB cosϕ = sin 2 ϕ When ϕ = π 2 ⇒ cosϕ = 0 sinϕ = 1 y 2 B 2 + x 2 A 2 = 1 . . . . ( 3 )
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