A tube of diameter $d$ is lowered into a vessel with water. A ball of the same diameter is placed in the tube. The center of the ball is at a depth $h$. Find the density of the substance of the ball. There is no gap between the pipe and the ball. The frictional force between them is zero.

Two identical bathyspheres float in suspension, the first at depth $2d$, the second at depth $d$ ($d = 1 km$). At the initial moment of time the first bathysphere discharges ballast and starts to float. When it floats to depth d, the second bathysphere bathysphere discharges ballast and begins to float. The first bathysphere appeared on the water surface at $T = 10 s$ earlier than the second one. It is known that the first bathysphere moved with almost constant velocity $v_{0} = 1 m/s$ for the second half of its path. Find the ejective force. The mass of the bathysphere without ballast is $m$. (The force of resistance to the motion of the bathysphere from the water side can be considered directly proportional to the speed of the bathysphere.

Lead and aluminum balls of equal radius $r$ are connected by a weightless non-stretchable thread, the length of which is much greater than the size of the balls. The balls were lowered in a vessel with glycerol. They then came to motion with zero initial velocity. The drag force on the balls is proportional to their velocity, and the coefficient of proportionality is the same for both balls. Find the tension force of the thread at the steady-state velocity of the balls. The densities of aluminum and lead are $\rho_{1}$ and $\rho_{2}$.

A beaker of mass $m = 100 g$ and volume $V = 200 cm^{3}$ is suspended upside down from a dynamometer, with the edges of the beaker touching the surface of water and the inside volume of the beaker filled with water. Why does the water not pour out of the beaker? What does the dynamometer show?

A hermetically closed absolutely rigid vessel of height $H$ is filled to the top with water. The water pressure at the upper wall of the vessel is zero. At the bottom of the vessel there is a small air bubble. How will the pressure of the water in the vessel change if the bubble floats up? Assume that water is incompressible. Neglect the solubility of air in water.

The king ordered Archimedes to measure as much gold as an elephant weighs. Such large scales were nowhere to be found. Archimedes had a raft at his disposal. How did Archimedes solve the problem?

A body completely immersed in a liquid floats at some depth in equilibrium. What physical parameters determine the stability of this equilibrium? Write down the condition of stability of equilibrium, considering that the temperature of the liquid does not depend on the depth.

Two metal balls (a lead ball and an iron ball) are balanced on different weights. When the balls are lowered into water, the equilibrium is not disturbed. Try to find an explanation for this phenomenon.

An ice flake with a cork (pellet) frozen into it is floating in a glass of water at a temperature of $0^{ \circ} C$. How will the water level in the beaker change if the ice melts and the temperature remains the same?

In a large vessel with water floats in a vertical position a thin-walled beaker in which a certain amount of water is poured. The difference between the levels of water in the vessel and the beaker is $x$. How will this difference change if a cork is put into the beaker?

A thick-walled boat with vertical walls and a hole in the bottom floats freely in the lake for quite a long time. Then the hole is plugged from the outside and a log is lowered inside the boat. Will the water level in the boat rise or fall after this relative to the water level in the lake? Why?

A washer of mass $M$, having the form of a cylinder with the area of the base $S$ and height $h$, floats on the interface of two immiscible liquids with densities $\rho_{1}$ and $\rho_{2}$ ($\rho_{1} < \rho_{2}$).The base of the washer is parallel to the liquid interface.Find the depth of immersion of the washer in the lower fluid.

A cube floats in a vessel with water so that its upper edge is parallel to the water surface.Half of the cube is immersed in the water.What layer of oil must be added to make the cube float completely immersed in the liquid, if the density of oil is half that of water and the length of the cube's edge is $a$? Oil and water do not mix.

The weight $P$ of a system consisting of a beaker of water and a cork ball is measured in five cases:
1) the ball floats freely in the beaker (scale reading $P_{1}$);
2) the ball is lying on the scale next to the beaker ($P_{2}$);
3) the ball is held fully submerged by a thin weightless string attached to the bottom of the beaker ($P_{z}$);
4) the ball is held in a fully submerged state by a thin weightless spoke attached above the beaker ($P_{4}$);
5) the ball held in a drowned state is released and begins to float freely ($P_{5}$$).
Arrange the readings of the scales in ascending order.

A weight and a glass of water stand on a scale. Compare the reading of the scale $P$ with the reading of the weights in the following cases:
1) the kettlebell is suspended by a thread from an external tripod and is submerged in water ($P_{1}$);
2) the suspension is broken and the kettlebell begins to sink ($P_{2}$).