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Suppose I have 2 fluids of different densities, and I pour the same mass of each fluid into 2 identical cups. I feel like I'm confusing myself here — on one hand hand, both cups contain the same mass of fluid, so mg/A is equal for both. At the same time, the denser liquid occupies less volume, and the depth from the atmosphere to the bottom of the cup is less. Am I being too naive and forgetting something, like omni-directionality or something along those lines? Does the difference is density perfectly match out the difference in depth? Thank you!

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    $\begingroup$ Hydrostatic pressure is calculated via rhogh. The heigher liquid column of the less dense fluid makes up for the lower density. Set values for both fluids and make the calculations. Also you might want to look up the hydrostatic paradox. $\endgroup$ – idkfa Apr 24 '17 at 6:13
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You are correct. The difference in density matches perfectly depth occupied in a container of constant horizontal cross section. Refer to buoyancy as it works similarly - the depth a less dense liquid in a container that is floating sinks in the liquid the container floats in is related to the densites of the 2 liquids.

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  • $\begingroup$ Hello, thank you for the answer! Sorry but I can't exactly see how this relates to buoyancy, they feel too distinct to me for me to make a connection. Could you elaborate more please? $\endgroup$ – user107224 Apr 27 '17 at 12:40
  • $\begingroup$ You do not understand perhaps because you are not thinking on a simple enough level. The link between your situation and buoyancy is that the basis for both is density. The buoyancy (floatability) of something is a function of the density - the less dense the volume of the mass the more buoyant. And the ratio of densities of the vessel to the water determines how much of the vessel floats. $\endgroup$ – john Apr 28 '17 at 20:00
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It depends on the shape of the container. If the sides are vertical, then the ratio of depths will be inversely as the ratio of densities, and the pressure at the bottom of each will be the same: in one container you have a larger depth, but a smaller pressure gradient, in the other a smaller depth but an equally larger pressure gradient. However, if the sides of the containers slope, then all bets are off, because the depth no longer scales proportionally with the volume.

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It may help you understand what is happening here if you think in terms of weight rather than in terms of mass. Weight and mass are related by the equation $W = mG$, where $W$ is the weight of an object, $m$ is the mass of the object, and $G$ is the local gravitational constant ($32.2 \text{ ft/sec/sec}$ at sea level). If your 2 identical cups are at about the same elevation, $G$ will be essentially the same for both cups. Therefore, you can weigh and put in say 2 lbs of the less dense fluid into cup 1, and 2 lbs of the heaverier fluid in cup 2. Assuming your cups have vertical side walls, cup 1 will be more full than cup 2.
Since density, $d$, is $W$/volume, $A \cdot h$ where $A$ is the area of the cup cross section and $h$ is the height of the fluid, we have:

cup 1 $$h_1 A d_1 = W_1 = 2 \text{ lbs}$$ cup 2 $$h_2 A d_2 = W_2 = 2 \text{ lbs}$$

and we get $h_1 d_1 = h_2 d_2$, or $h_1/h_2 =d_2/d_1$, so yes, the ratio of the heights is inversely proportional to the ratio of the densities.

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