A V-shaped glass tube of uniform cross section is kept in a vertical plane as shown. A liquid is poured in the tube. In equilibrum the level of liquid in both limds of tube. Find the angular frequency of small oscillation of liquid.

Explanation

Let us displace the liquid of left limb by a distance x, the liquid of right limb will rise relative equilibrum line by same distance x

Excess length of the liquid on the right limb
`Delta l = x + x`
(i)
From the figure ,
`y = x sin alpha = x' sin beta`
(ii)
from eq. (i)
`x = (x sin alpha)/(sin beta)`
from eqs (i) and (iii) , `Delta l = ((sin alpha + sin beta)/(sin beta)) alpha`
This excess length `Delta l` in right limb will proved restoring force to the liquid in tube
Weight of the liquid column
`W = p A. Delta l s = ((sin alpha + sin beta)/(sin beta)) rho Axg` ltbr `rho` is the density of liquid and A is the area of cross section of tube. The component of `W` along the tube will be equal to restoring force
`F = W sin beta = rho (sin alpha + sin beta)Axg`
This force helps the entire liquid to restore its original position by moving it which an acceleration `a = F//m, m = mass` of total liquids
`a = (rho (sin alpha + sin beta) Axg)/(m)`
Here `m =` mass of liquid in the tube `= A rho l l = `Length of total liquid column
`= (h)/(sin alpha) + (h)/( sin beta) = h [(sin alpha + sin beta)/(sin alpha + sin beta)]`
Hence `a = (rho (sin alpha + sin beta) Axg)/(A rho [h (sin alpha + sin beta)/(sin alpha. sin beta)])`
`bara = - ((g sin alpha. sin beta)/(h)) barx [("with proper"),("vector sign")]`
Compare with `bara = - omega^(2) bar x`
which gives
`omega = sqrt((g sin alpha . sin beta)/(h)) implies T = 2 pi sqrt((h)/(g sin alpha . sin beta))`