A tube of length 1 m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is √F/(α M) in SI unit. The value of α is ______.

Idea: the liquid is spread along the tube, so no single radius applies. Integrate the centripetal requirement, or equivalently put the whole mass at the centre of mass.

With mass per unit length (2M)/L, the force at the far end must supply the centripetal force for every element:

F=∫₀^L(2M)/Lω²r dr=(2Mω²)/L·(L²)/2=Mω²L.

That is the same as putting the whole 2M at the centre of mass, L/2 from the axis: 2Mω²L/2=Mω²L.

With L=1 m,

F=Mω², so ω=√F/M.

Comparing with √F/(α M) gives α=1.

Treating all 2M as sitting at the far end would give F=2Mω²L and hence α=2; it is the factor of two from the centre of mass that brings it down to exactly 1.