\({t_{Ac}} = {t_{AB}} + {t_{BC}}\)

\( = (T/4) + {t_{BC}}\)

where \(T =\) time period of oscillation of spring mass system

\({t_{BC}}\) can be obtained from, \(BC = AB\sin (2\pi /T)\,{t_{BC}}\)

Putting \(\frac{{BC}}{{AB}} = \frac{1}{2}\) we obtain \({t_{BC}} = \frac{T}{{12}}\)

==> \({t_{AC}} = \frac{T}{4} + \frac{T}{{12}} = \frac{{2\pi }}{3}\sqrt {\frac{m}{k}} \).