Question

The displacement of a particle is represented by the equation y = sin³ωt. The motion is ______.

विकल्प

• non-periodic.

• periodic but not simple harmonic.

• simple harmonic with period 2π/ω.

• simple harmonic with period π/ω.

Answer 1

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.