# oil and water- The Archimides' law

I have a problem as on the topic, see the below.

First I added $25\hbox{cm}^3$ water in to a beaker. Then I added $2N$ oil onto it. But the level of the water was still $25\hbox{cm}^3$.

My problem is this. Archimedes' principle states that,

the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid.

So according to this situation the upward buoyant force by water should equal to the the weight of the oil due to the equilibrium. Also it should be same to the weight of the displaced water volume. But as the practical shows the displaced water volume is $0$.So the upward buoyant force should be 0. But according to the equilibrium the upward buoyant should be $2N$.

Where is the mistake?

## 1 Answer

Archimedes' principle isn't directly applicable here, since there's nowhere for the water to be displaced to. If you filled a thinner, weightless beaker with the oil and tossed that into the original beaker, you will indeed see water displaced. But in your example, the water can't even be displaced (where would it go?) so you don't see displacement. The entire system of oil and water can be treated as one.

Here's another way to think about it. In the figure on the left, imagine cutting the water in half such that the top layer is resting on the bottom one. The top layer is clearly in equilibrium, or it would move. But gravity is acting on it, so there must be a force cancelling gravity. The bottom layer isn't displaced either, so buoyancy isn't the source of this force. Instead it's coming from the internal pressure of water, which is preventing the bottom layer from being compressed.