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  (Fig.). 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 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 equilibrium, neglecting friction.

water are $0.5\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3},2.7\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$, and $1\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ respectively.

Determine the minimum mass $m_{\mathrm{al} }$ of the wire that should be wound on the cork so that the cork with the wire is completely submerge $1$ in water.

One end of an iron chain is fixed to a sphere of mass $M=10 \mathrm{kg}$ and of diameter $D=0.3 \mathrm{m}$ (the volume of such a sphere is $V=0.0141{ \mathrm{m}}^3$, while the other end is free. The length $l$ of the chain is $3 \mathrm{m}$ and its mass $m$ is $9 \mathrm{kg}$. The sphere with the chain is in a reservoir whose depth $H=3 \mathrm{m}$.

Determine the depth at which the sphere will float, assuming that iron is $7.85$ times heavier than water.

Two bodies of the same volume but of different masses are in equilibrium on a lever.

Will the equilibrium be violated if the lever is immersed in water so that the bodies are completely submerged?

A flat wide and a high narrow box float in two identical vessels filled with water. The boxes do not sink when two identical heavy bodies of mass $m$ each are placed into them.

In which vessel will the level of water be higher?

A steel ball floats in a vessel with mercury.

How will the volume of the part of the ball submerged in mercury change if a layer of water completely covering the ball is poured above the mercury?

A piece of ice floats in a vessel with water above which a layer of a lighter oil is poured.

How will the level of the interface change after the whole of ice melts? What will be the change in the total level of liquid in the vessel?

A homogeneous aluminium ball of radius $r=0.5 \mathrm{cm}$ is suspended on a weightless thread from an end of a homogeneous rod of mass $M=4.4 \mathrm{g}$. The rod is placed on the edge of a tumbler with water so that half of the ball is submerged in water when 

the system is in equilibrium (Fig.). The densities ${\rho }_{\mathrm{a}1}$ and ${\rho }_{\mathrm{w}}$ of aluminium and water are $2.7\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$ and $1\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-9}$ respectively.

Determine the ratio $y/x$ of the parts of the rod to the brim, neglecting the surface tension on the boundaries between the ball and water.