Question

Two cylinders with a horizontal and a vertical axis respectively rest on a horizontal surface. The cylinders are connected at the lower parts through a thin tube. The horizontal cylinder of radius r is open at one end and has a piston in it. The vertical cylinder is open at the top. The cylinders contain water which completely fills the part of the horizontal cylinder behind the piston and it is at a certain level h in the vertical cylinder.

Answer 1

Answer:

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Answer 2

Apply Pascal's Law

Step-by-step explanation:

The pressure at the bottom of the "vertical" cylinder is  

$p = p_0 + \rho_w gh$

where $p_0$ is the atmospheric pressure, ρw is the density of water, and g is the free-fall acceleration.

According to Pascal's law, the same pressure is exerted on the lower part of the piston in the "horizontal" cylinder. The total pressure of water on the part of the piston separated from the lower part by a distance x along the vertical is $p - \rho gx$.

Let us consider the parts of the piston in the form of narrow (of width Δx) horizontal strips separated by equal distances a from its center. The force of pressure exerted by water on the upper strip is

$[p - \rho_wg (r+a)] \Delta S$,

while the force of pressure on the lower strip is

$[p - \rho_wg (r-a)]\Delta S$,

where \delta S is the area of a strip. The sum of these forces is proportional to the area of the strip, the proportionality factor $2 (p - \rho gr)$ being independent of a. Hence it follows that the total force of pressure of water on the piston is

$(p - p_w gr)\pi r^2 = [p_0 + \rho_wg(h-r)]\pi r^2$

The piston is in equilibrium when this force is equal to the force of atmospheric pressure acting on the piston from the left and equal to $p_0 \pi r^2$.

Hence h = r, i.e. the piston is in equilibrium when the level of water in the vertical cylinder is equal to the radius of the horizontal cylinder.

An analysis of the solution shows that this. equilibrium is stable.