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Question
What is the ratio of maxmimum acceleration to the maximum velocity of a simple harmonic oscillator?
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Solution
Let x = A sin ω is the displacement function of SHM.
Velocity, `v = (dx)/(dt) = Aω cosωt`
`v_max = Aω |cos ωt|_"max"`
= `Aω xx 1`
= `ωA` [∵ |cos ωt|max = 1] ......(i)
Acceleration, `a = (dv)/(dt) = - ωA * ωsinωt`
= `- ω^2A sinωt`
`|a_max| = |(- ω^2A)(+1)|` ......[∵ (sin ωt)max = 1]
`|a_max| = ω^2A` .......(ii)
From equations (i) and (ii), we get
`v_max/a_max = (ωA)/(ω^2A) = 1/ω`
⇒ `a_max/v_max = ω`
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