Question

A vertical U-tube of uniform area of cross-section contains water upto a height of 60 cm. If the water (ρ_w = 1000 Kg/m³ ) is depressed on the surface and then released such that the water level moves up and down in both the limbs. Show that the up and down motion of water level is simple hormonic and find its time period. (g = 9.8 m s^–2)

Answer 1

Let water level be initially at the positions A and B at a height of 60 cm in the U-tube. If the water level is depressed through a distance y in one of limbs from A to A', water rises in the second limb through y from B to B' so that the difference of level in two limbs, A'B', is 2y.

Pressure difference in two limbs = difference in levels × density of water × g = 2y × ρ_w × g

Restoring force on liquid = 2y × ρ_w × g × S

where S is area of cross-section of tube.

If m is mass of liquid of height h in each limb initially, m = 2hS × ρ_w

If a is acceleration imparted to mass m of liquid due to restoring force, a

As acceleration a is directly proportional to vertical displacement and is directed towards the mean positions A or B, the water level executes S.H.M. Its time period T is given by: