Question

A tube filled with water and closed at both ends uniformly rotates in a horizontal plane about the OO'-axis. The manometers fixed in the tube wall at distances r₁ and r₂ from the rotational axis indicate pressures p₁ and p₂ respectively (Fig. 56).

Determine the angular velocity co of rotation of the tube, assuming that the density ρ_w of water is known.

Answer 1

Let us consider the conditions of equilibrium for the mass of water contained between cross sections separated by x and x + Δx from the rotational axis relative to the tube. This part of the liquid, whose mass is ρ_wS Δx, uniformly rotates at an angular velocity ω under the action of the forces of pressure on its lateral surfaces. Denoting the pressure in the section x by p (x), we obtain

Making Δx tend to zero, we obtain the following equation:

Using the conditions of the problem

we obtain the angular velocity of the tube: