In an oscillating system, the restoring force F is defined as,
F = kx
Here k is the force constant and x is the displacement of the particle from its mean position.
The restoring force F acting on the liquid is defined as,
F = 2ρAxg
Here ρ is the density of the liquid, A is the area of cross section, x is the height from equilibrium position and g is the acceleration of free fall.
If we compare the equation F = kx with F = 2ρAxg, the force constant k will be,
k = 2ρAg
Density (ρ) is equal to mass (m) per unit volume (V).
ρ = m/ V
So using equation ρ = m/ V, mass m of the fluid in the U-tube having area A and total length of the liquid L will be,
ρ = m/ V
m= ρ V
= ρ(AL) (Since V= AL)
To find out the period of oscillation T, substitute ρ(AL) for m and 2ρAg for k in the equation T = 2π √m/k,
T = 2π √m/k
= 2π √( ρ(AL) /(2ρAg)
= π√2L/g
From the above observation we conclude that, period of oscillation will be π√2L/g.