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Question
The displacement of a particle is represented by the equation y = sin3ωt. The motion is ______.
Options
non-periodic.
periodic but not simple harmonic.
simple harmonic with period 2π/ω.
simple harmonic with period π/ω.
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Solution
The displacement of a particle is represented by the equation y = sin3ωt. The motion is periodic but not simple harmonic.
Explanation:
Given the equation of motion is y = sin3ωt
= `(3 sin ωt - 4 sin 3 ωt)/4` .....[∵ sin 3θ = 3 sin θ – 4sin3θ]
⇒ `(dy)/(dt) = ([d/(dt) (3sin ωt) - d/(dt) (4sin3ωt)])/4`
⇒ `4 (dy)/(dt) = 3ωcos ωt - 4 xx [3ωcos 3ωt]`
⇒ `4 xx (d^2y)/(dt^2) = - 3ω^2sin ωt + 12 ωsin3ωt`
⇒ `(d^2y)/(dt^2) = - (3ω^2 sinωt + 12ω^2 sin 3ωt)/4`
⇒ `(d^2y)/(dt^2)` is not proportional to y.
Hence, the motion is not SHM.
As the expression is involved in function, hence it will be periodic.
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