What is the ratio of maximum acceleration to the maximum velocity of a simple harmonic oscillator?

Explanation

Let `x=sinomegat` is the displacement function of SHM.
Velocity, `v=(dx)/(dt)=Aomegacosomegat`
`v_(max)=Aomega|cosomegat|_(max)`
`=Aomega xx1=omegaA` `[:'(cosomegat)_(max)=1]......(i)`
Acceleration, `a=(dv)/(dt)=-omegaA.omegatsinomegat`
`=-omega^(2)Asinomegat`
`|a_(max)|=|(-omega^(2)A)(+1)|` `[:'(sinomegat)_(max)=1]`
`|a_(max)|omega^(2)A" "......(ii)`
From Eqs. (i) and (ii), we get
`(v_(max))/(a_(max))=(omegaA)/omega^(2)A=(1)/(omega)`
`rArr(a_(max))/v_(max)=omega`