Final answer:
The force exerted by the liquid at the opposite end when a tube filled with an incompressible fluid is rotated about one end is MLω²/2, implying that the correct answer is option A.
Step-by-step explanation:
The force exerted by the liquid at the other end of a rotating tube filled with an incompressible fluid, when rotated in a horizontal plane about one of its ends, can be found by considering the liquid's circular motion. In such motion, every particle of the liquid will experience a centripetal force necessary to keep it in a circular path, which in turn is provided by the pressure exerted on the walls of the tube.
The mass M of the liquid at a distance r from the axis is experiencing centripetal acceleration rω², and so the force F on the liquid element of mass dm is dF = dmω²r. Upon integrating across the length of the tube, we find that the force exerted by the liquid at the opposite end is F = MLω²/2, which corresponds to option A.