Question

A U–tube of uniform cross–section contains a liquid of density ρ up to a height h in each limb. The tube is rotated with a constant angular speed ω about one limb as axis as shown in Fig. The difference in the level of the liquid in the two limbs of U–tube is

(1) ω²L/g

(2) ω²L²/g

(3) ω²L²/2g

(4) ωL²/2g²

Answer 1

(3) ω²L²/2g

Consider a small element LM of liquid defined by x and x+dx as shown in Fig.

dm = The mass of element considered = (Adx)_p

where A is area of cross–section of limb of U–tube. The element LM moves in a circular path of radius x. It experience centrifugal force dF. Obviously

dF = (dm)xω² = Apω² xdx

Let P₁ and P₂ be liquid pressure at bottom of left and right hand limb. Then

Let h be the difference in level of liquid in the two limbs. Then

P₂ - P₁ = hpg ....(ii)

From Eqns (i) and (ii) we have

h = ω²L²/2g