Fluid, pressure and weight question

[fluidistic]

Hi PF,
I think I solved well the problem but I don't know what's going on and I thought about it last night and I am at a loss on this...

Homework Statement

Consider a closed cylinder whose height is H and with radius R. On its top there is a cylinder of height H and radius $$\frac{R}{100}$$ such that we can fill the 2 cylinders with liquid via the cylinder on the top of the other. (They are connected such that the liquid can flow between the cylinders).
Compare the weight of the liquid with the hydrostatic force supported by the base of the first cylinder.

Homework Equations

$$\Delta F=P \Delta A$$
$$P=\rho gH$$

The Attempt at a Solution

I found out that the weight of the liquid is $$\frac{5001 \rho \pi R^2 gH}{5000}$$ while the force supported by the base of the first cylinder is $$\frac{5000.5 \rho \pi R^2 gH}{5000}$$. I find this incredible that the ground doesn't have to support all the weight of the liquid... I don't know what's going on. The problem is of course under the column of water formed by the second cylinder. It seems that its weight is greater than the force needed to support it. Maybe molecules are going upward there and so forming a flow... I really want to know!

 

[Nate810]

The answer is that the hydrostatic force supported by the base of the first cylinder is greater than the weight of the liquid. This is because the pressure on the base of the first cylinder is increased by the column of water formed by the second cylinder. The pressure on the base of the first cylinder is the sum of the pressure due to the weight of the liquid in both cylinders plus the pressure due to the column of water formed by the second cylinder. This extra pressure increases the hydrostatic force that the base of the first cylinder must support, making it greater than the weight of the liquid.

 

[nlightner]

Hello,

Your solution seems to be correct. The reason why the weight of the liquid is greater than the hydrostatic force supported by the base of the first cylinder is due to the pressure at the bottom of the second cylinder. As the liquid is being filled into the two cylinders, the pressure at the bottom of the second cylinder increases, creating a force that is greater than the hydrostatic force. This is because the pressure at the bottom of the second cylinder is not just due to the weight of the liquid above it, but also the weight of the liquid in the first cylinder. This creates a net force that is greater than the hydrostatic force at the base of the first cylinder.

In addition, the flow of molecules you mentioned is also a contributing factor to the difference in weight and force. As the liquid is being filled, there is a movement of molecules from the top of the second cylinder to the bottom, creating a flow and increasing the pressure at the bottom. This flow also contributes to the net force at the base of the first cylinder.

Overall, the weight of the liquid is greater than the hydrostatic force due to the pressure and flow of molecules in the system. I hope this helps clarify the situation. Keep up the good work in solving problems and questioning your understanding!