This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

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Explanation for dam walls: Liquid pressure increases with depth. The water at the bottom of the dam exerts a much greater pressure on the walls than the water near the surface. To withstand this higher pressure and prevent the dam from collapsing, the walls are built thicker at the bottom.
a) Observation when thistle funnel is moved downwards: The difference in the liquid levels in arms A and B of the U-tube manometer will increase. This is because liquid pressure increases with depth. As the thistle funnel is moved downwards, the depth of immersion increases, leading to a greater pressure exerted on the membrane of the funnel. This increased pressure is transmitted through the air in the tube to the manometer, causing a larger difference in the liquid levels.
b) Observation when water is replaced with glycerine at the same depth: The difference in the liquid levels in arms A and B of the U-tube manometer will increase compared to when water was used. This is because liquid pressure also depends on the density of the liquid. Glycerine is denser than water. Since pressure , and the depth is kept constant, a denser liquid () will exert a greater pressure, resulting in a larger difference in the manometer levels.
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This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.