One end of a V-tube containing mercury is connected ot a suction pump and the other end to atmosphere. The two arms of the tube are inclined to horizontal at an angle of `45^(@)` each. A small pressure difference is created between two columns when the suction pump is removed. Will Neglect capillary and viscous forces. Find the time period of oscillation.

Explanation

Consider the diagram shown below

Let us consider an infinitesimal liquid column of length dx at a height x from the horizontal line.
If p = density of the liquid
A= cross-sectional area of V-tube
PE of element dx will be given as
`PE=dmgx=(Apdx)gx" " [:.dm=pv=pAdx]`
where Apdx=dm=mass of element dx
Total PE of the left column
`underset(o)overset(h_(1))intApgdx`
`=Apgunderset(o)overset(h_(1))intxdx`
`=Apg|(x^(2))/(2)|^(h_(1))=Apgh_(1)^(2)/(2)`
But, `h_(1)=lsin45^(@)`
`:.PE=(Apg)/(2)l^(2)sin^(2)45^(@)" .....(i)`
In a similar way,
`"PE of right column"=(Apg)/(2)l^(2)sin^(2)45^(@)" ".....(ii)`
Total `PE=(Apg)/(2)l^(2)sin^(2)45^(@)+(Apg)/(2)l^(2)sin^(2)45^(@)`
`=2xx(1)/(2)Apgl^(2)((1)/(sqrt2))^(2)=(Apgl^(2))/(2)" " ....(iii)`
If due to pressure difference created y element of left side moves on the side, then
liquid present in the left arm=l-y
liquid present in the right arm=l+y
`:."Total PE"=Apg(l-y)^(2)sin^(2)45^(@)+Apg(l+y)^(2)sin^(2)45^(@)`
`"Changes in PE"=(PE)_("final")-(PE)_("initial")`
or `DeltaPE=(Apg)/(g)[(l-y)^(2)+(l+y)^(2)-l^(2)]`
`=(Apg)/(2)[l^(2)+y^(2)-2ly+l^(2)+y^(2)+2ly-l^(2)]`
`=(Apg)/(2)[2(l^(2)+y^(2))]`
`=Apg(l^(2)+y^(2))" " ....(iv)`
If v is the change in velocity of the total liquid column, then change in KE
`DeltaKE=(1)/(2)mv^(2)`
But `m=Ap(2l)`
`DeltaKE=(1)/(2)Apg2lv^(2)=Aplv^(2)" ".....(v)`
From Eqs. (iv)and (v),
`DeltaPE+KP=Apg(l^(2)+y^(2))+Aplv^(2)" ".....(vi)`
Sytem being conservative.
:. Change in total enregy =0
From Eq.(vi)
`Apg(l^(2)+y^(2))+Aplv^(2)=0`
Differentiating both sides with respect to time (t), we get
`Apg[0+2y(dy)/(dt)]+Apl(2v)(dv)/(dt)=0`
But `(dy)/(dt)=vand (dv)/(dt)=a" "["accelecration"]`
`rArrApg(2u=yv)+Ap(2v)=0`
`rArr(gy+la)2Apv=0`
`"2Apv=constant" and 2Apvne0`
`:.la+gy=0`
`a+((g)/(l))y=0`
or `(d^(2)y)/(dt^(2))+((g)/(l))y=0`
This is the standard differential equation for SHM of the form
`(d^(2)y)/(dt^(2))+omega^(2)y=0`
`:.omega=sqrt((g)/(l))`
`:.T =(2pi)/(omega)=2pisqrt((l)/(g))`