Question

A particle moves on the X-axis according to the equation x=x₀sin²⍵t. The motion is simple harmonic
(a) with amplitude x₀
(b) with amplitude 2x₀
(c) with time period 2π/⍵
(d) with time period π/⍵

Answer 1

Since, it is mentioned that the motion is Simple Harmonic, so we need to bring this into the Mathematical Form of the Simple Harmonic Motion.

∴ x = x₀sin²wt

Cos2θ = 1 - 2Sin²θ      [Let θ = wt.]

∴ Sin²θ = (1 - Cos2θ)/2

∴ x = x₀(/2 - Cos2θ/2)

x = x₀/2 - x₀Cos2θ/2

x - x₀/2 = -x₀/2 Cos2θ

x - x₀/2 = -x₀/2 Cos2wt

Therefore, comparing with Standard equation,

y = ASin(wt + Φ), therefore, A = -x₀/2.

Thus, there is no options like this. hence, (a). and (b) are incorrect options.

Now, Time Period(T) = 2π/ω'

In this case, ω' = 2ω

∴ T = 2π/2ω

∴ T = π/ω

Therefor,e Option (d). is correct.

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Hope it helps.