Pressure with depth formula derivation

Question

Consider a unit volume element of fluid consisting two horizontal circular areas S1 and s2, connected by vertical lines as boundries.

(sorry,I can't upload an image)

Now for derivation of pressure change with depth we need to make an fbd for the above described unit volume element.

For this my book takes weight downward , F1 on the bottom surface S1 upwards and F2 on the top surface S2 downwards.

My question is won't there be a force downwards on S1 and upwards on S2, which will have an effect of cancelling out F1 and F2 respectively.

I have reasoned this out from the fact that pressure is same in all directions in a fluid. Please help me clear this doubt.

Thanking you in advance.

Answer 1

won't there be a force downwards on S1 and upwards on S2, which will have an effect of cancelling out F1 and F2 respectively.

Imagine that the described parcel of water is a single object, disconnected from the rest of the water. You can pretend that it is replaced with a solid container of the same shape and mass.

A free body diagram shows the forces that apply to an object. Yes, the object will create reaction forces in the opposite direction (Newton's third law), but we will not concern ourselves with them.

Now the only vertical forces on this object are the water pushing from below, the water pushing from above, and gravity pulling it down. S1 should be seen as a description of the boundary for this object, not a separate object that will be pushed.