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

Options

A. `T=2pi sqrt((MA)/(gd))`
B. `T=2pi sqrt((M)/(2Adg))`
C. `T = 2pi sqrt((M)/(g))`
D. `T=2pisqrt((M)/(gdA))`

Answer

B

Explanation

When the liquid in the left arm is depressed by y cm, the liquid in the right arm is lifted through y cm and thus there is a level difference of 2y cm.
`therefore` The weight of liquid column of height 2y is
= mg =vdg=A (2y) dg and it acts as a restoring force on the entire mass M of the liquid.
`therefore` Restoring acceleration `a= -(2Aydg)/(M) therefore a prop (-y)`
`therefore` Thus it executes a SHM of `T=2pisqrt(("displacement")/("acceleration"))`
`therefore T=2pi sqrt((y)/(2Aydg//M))=2pi sqrt((M)/(2Adg))`