The minimum time taken by a spring block system (having time period T) to travel a distance equal to amplitude of motion is equal to

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Velocity-time graph of a body executing S.H.M. is

So to cover a given distance in minimum time velocity should be greater. So it must be around mean position and minimum time taken to cover a distance $\frac{a}{2}$ is $\frac{T}{12}$

$∴ T_{R e q u i r e d} = \frac{T}{6}$

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