A tube of length 50 cm is filled completely with an incompressible liquid of mass 250 g and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity x√F rad s⁻¹. If F is the force exerted by the liquid at the other end, then the value…

Question

A tube of length 50 cm is filled completely with an incompressible liquid of mass 250 g and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity x√F rad s⁻¹. If F is the force exerted by the liquid at the other end, then the value of x will be .

Accepted Answer

4. Given: L=50 cm=0.5 m, m=250 g=0.25 kg, rotation about one end in a horizontal plane, so gravity plays no part. Take an element of length dr at distance r from the axis; its mass is m/Ldr and it needs a centripetal force m/Lω²r dr. The pressure at the axis end is atmospheric, so the whole of this is supplied by the far end: F=∫₀^Lm/Lω²r dr=(mω²L)/2. F=(0.25×0.5)/2ω²=(ω²)/(16), so ω=4√F. Comparing with ω=x√F gives x=4.

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