Final answer:
The force exerted on the bases of two vessels with the same base area and filled water to the same height is the same, due to Pascal's law. The difference in weight readings is due to the different volumes the vessels can hold based on their shape, which impacts the total weight of the water each can contain.
Step-by-step explanation:
The force exerted by the water on the base of two vessels with the same base area but different volumes to a common height is indeed the same. The force exerted on the base, according to Pascal's law, is due to the weight of the water directly above it and the pressure exerted by the water. As the base areas are identical and the water height is the same, the pressure at the base of both vessels is equal, leading to the same force exerted on both vessels, regardless of their shape or volume.
The differing readings on a weighing scale come from the fact that although the vessels are filled to the same height, the first vessel holds twice the volume of water, and therefore twice the weight, compared to the second vessel. This is because the first vessel likely has a wider shape that expands as it goes upwards, or maybe it branches out in some form, which allows it to contain more water compared to the second vessel that may have a constant or narrowing shape.