Simple Harmonic Motion

A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency \(\omega\) and amplitude \(a\). If at a height \(y^{\prime}\) from the mean position, the body gets detached from the spring, calculate the value of \(y^{\prime}\) so that the height \(\mathrm{H}\) attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment \(\left(a \omega^{2}>g\right)\)