Question

A U tube is rotated about one of its limbs with a n angular velocity $\omega$. Find the difference in height H of the liquid (density $\rho$) level, where diameter of the tube d<<L.

Answer 1

Weight of liquid of height H

= πd² / 4 x H x ρ x g …(i)

Let us consider a mass dm situated at a distance x from A as shown in the figure. The centripetal force required for the mass to rotate = (dm) xω²

∴ The total centripetal force required for the mass of length L to rotate

= xω² where dm = ρ x πd² / 4 x dx

∴ Total centripetal force

= x (xω²)

= ρ x πd² / 4 x ω² x L² / 2 …(ii)

This centripetal force is provided by the weight of liquid of height H.

From (i) and (ii)

πd² / 4 x H x ρ x g = ρ x πd² / 4 x ω² x L² / 2

H = ω² L² / 2g