A simple harmonic oscillator of angular frequency 2 rad s - 1 is acted upon by an external force F = sin ⁡ t N . If the oscillator is at rest in its equilibrium position at t = 0 , its position at later times is proportional to:

Question

A simple harmonic oscillator of angular frequency 2 rad s - 1 is acted upon by an external force F = sin ⁡ t N . If the oscillator is at rest in its equilibrium position at t = 0 , its position at later times is proportional to:, sin ⁡ t + 1 2 cos ⁡ 2 t, c o s t - 1 2 sin ⁡ 2 t, sin ⁡ t - 1 2 sin ⁡ 2 t, sin ⁡ t + 1 2 sin ⁡ 2 t

MARKS App · Accepted Answer

It is given that oscillator at rest at t = 0 i.e. at t = 0 , v = 0 . So, in option, we can check by putting v = d x d t = 0 . (1) I f x ∝ sin ⁡ t + 1 2 cos ⁡ 2 t ⟹ v ∝ cos ⁡ t + 1 2 × 2 ( - sin ⁡ 2 t ) ⟹ a t t = 0 , v ∝ 1 - 0 ≠ 0 (2) I f x ∝ cos ⁡ t - 1 2 sin ⁡ t ⟹ v ∝ - sin ⁡ t - 1 2 cos ⁡ t ⟹ a t t = 0 , v ∝ - 1 2 ≠ 0 (3) I f x ∝ sin ⁡ t - 1 2 s i n 2 t t h e n υ ∝ cos ⁡ t - 1 2 × 2 cos ⁡ 2 t ⟹ a t t = 0 , v ∝ 1 - 1 = 0 (4) I f x ∝ sin ⁡ t + 1 2 sin ⁡ 2 t ⟹ v ∝ cos ⁡ t + 1 2 × 2 cos ⁡ 2 t ⟹ a t t = 0 , v ∝ 1 + 1 ⟹ v ∝ 2 ≠ 0 ∴ i n o p t i o n 3 v = 0 a t t = 0

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