59) A tube of length L is filled completely woth an incompressible liquid of mass M and closed at both the ends. The tube is then rotated in a horizontal plane about one of it's ends with a uniform angular velocity w. The force exerted by the liquid at the other end is (answer in terms og M, L and w(angular velocity))^{ }

According to the question, we have a tube of length L filled with liquid of mass M. As the tube is rotated in horizontal plane along a vertical axis passing through one of its ends with angular velocity ω, it exerts a force on the wall of the other end of the tube due to the centrifugal force experienced by it.

The mass per unit length of the tube is = M/L.

Now consider a small section of the tube ‘dx’ at a distance ‘x’ from the tube. The mass of liquid in this section is, dm = Mdx/L

The centrifugal force experienced by this mass is, dF = (dm)(ω^{2})(x)

As the mass of liquid extends from x = 0 to x = L, the net force on the other wall or the net centrifugal force experienced by the mass of the liquid is given by,

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