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 ($P_{2}$).

The cork is attached by a light spring to the bottom of the water vessel. Both the cork and the spring are completely immersed in the water. Will the length of the spring increase or decrease if the vessel moves vertically up (down) with constant acceleration?

A pipe of diameter $d$ is lowered into a vessel with water. A ball of the same diameter is placed in the pipe. 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 weightless springs have lengths $l_{1}, l_{2}$ and stiffnesses $k_{1}, k_{2}$. One spring is inserted into the other spring. The ends of the springs are connected in pairs. The other points of the springs do not touch each other. What is the stiffness $k$ of the resulting spring?

As you know, when calculating the period of oscillation of a mathematical pendulum, you substitute $\sin \alpha$ for $\alpha$, where $\alpha$ is the angle of deviation of the pendulum string from the vertical. Find out whether the period calculated in this way is smaller or larger than the real period. A mathematical pendulum is a point mass $m$ suspended on a weightless, inextensible thread of length $l$.

A small ball of mass m, attached to a spring with stiffness coefficient $k$, makes harmonic oscillations in the direction of the spring axis with amplitude $A$. When the spring was compressed, a second spring pendulum of the same mass is placed in the path of the ball, both springs having a common axis and the equilibrium positions of the balls coinciding. Describe the further motion of the system in the following cases:
(a) The shock is central and perfectly elastic;
b) the impact is central and absolutely inelastic (i.e. the balls "stick together").
At the initial moment the second ball rests in the equilibrium position.

In a deep vertical shaft underground there are two weights of equal mass connected by a weightless non-stretchable thread, which is thrown over a fixed block. Find the period of vertical oscillation of the weights if the radius of the Earth is 6400 km, the acceleration of free fall at the surface is $10 m/s^{2}$.

A projectile fired from a cannon mounted at an angle of $45^{\circ}$ to the horizon on a flat horizontal plain breaks at the upper point of its trajectory into two fragments of equal mass. The first fragment falls to the ground directly below the point of bursting 15 s later. At what distance from the cannon will the second fragment fall if 15 s elapsed from the moment of firing to the moment of rupture? Neglect air resistance.