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 $\left(Fig.53\right)$ Yhe "vertical" cylinder is open at the top. The cylinder contain water which completely fills the part of the horizontal cylinder behind the piston and is at a certain level in the vertical cylinder.Determine the level $h$ of water in the vertical cylinder at which the piston is in the equilibrium, neglecting friction.

Answer 1

The pressure at the bottom of the vertical cylinder is $p=p_0+{\rho }_{w}gh$, where $p_0$ is the atmospheric pressure, ${\rho }_w$ 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 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 $\Delta x$) horizontal strips separated by equal distances $a$ from its centre. The force of pressure exerted by water on the upper strip is $[p-\rho wg(r+a\left)\right]\Delta S$,
while the force of pressure on the lower strip is $[p-{\rho }_{w}g(r-a\left)\right]\Delta S$,
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
$(p-pwgr)\pi r^2=[p_0+{\rho }_{w}g(h-r\left)\right]\pi r^2$.
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 $p_0\pi r^3$. 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.