A tube of length L is filled completely with an incomeressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity `omega.` The force exerted by the liquid at the other end is

Explanation

As is know, in rotational motion, angular velocity `omega` remains constant for all segments of the liquid in the tube.
`:.` Force exerted by the liquid at the other end
=Force required for rotational motion
`F = mr omega^(2)` ….(i)
Consider a small element of the liquid of length `dx` at a distance `x` from the centre,
Mass of this element of the liquid
`m = (M)/(L) dx , r = x`
`:. dF = mr omega^(2) = ((M)/(L) dx) x omega^(2)`
`F = (M)/(L) omega^(2) int_(0)^(L) x dx = (M)/(L)omega^(2) [(x^(2))/(2)]_(0)^(L)`
`F = (M)/(L)omega^(2) ((L^(2))/(2)) = (1)/(2)M omega^(2)L`