A U - tube of uniform bore of cross-sectional area A has been set up vertically with open ends facing up. Now m mass of a liquid of density d is poured into it. The column of liquid in this tube will oscillate with a period T such that

If the level of liquid is depressed by y on one side, then the level of liquid in column P is 2y higher than other side as shown.
 

The weight of extra liquid on the side P:  $\mathrm{W}=2\mathrm{Aydg}$ 
This becomes the restoring force on mass M. 
Restoring acceleration, $\mathrm{a}=\frac{-2\mathrm{Aydg}}{\mathrm{M}}$ 
This relation satisfies the condition of SHM i.e.  

$\mathrm{\omega }=\sqrt{\frac{2\mathrm{Adg}}{\mathrm{M}}}$
Hence time period, 

$\mathrm{T}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$ 
$=2\mathrm{\pi }\sqrt{\frac{1}{\frac{2\mathrm{Adg}}{\mathrm{M}}}}\Rightarrow \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{M}}{2\mathrm{Adg}}}$