**P. 4821.** If a solid cork sphere of density 400 kg/m\(\displaystyle {}^3\) is suspended by a spring, then the elongation of the spring is 10 cm.

\(\displaystyle a)\) The sphere and the spring are both placed into a container which is filled with water, such that one end of the spring is attached to the bottom of the container, and the other is fixed to the cork sphere. The system is in equilibrium, the spring is stretched, and half of the cork sphere is immersed into the water. What is the elongation of the spring at this state?

\(\displaystyle b)\) What will the elongation of the spring be, if more water is poured into the container, such that the whole cork sphere is immersed into the water?

(4 points)

**P. 4822.** What should the angle between two forces be in order that their sum is equal to the

\(\displaystyle a)\) root-mean-square value of the magnitudes of the forces;

\(\displaystyle b)\) harmonic mean of the forces?

Under what conditions will these angles be minimal, and what are these minimum values?

(4 points)

**P. 4825.** A thermally insulated container is divided into two parts of volumes \(\displaystyle V\) and \(\displaystyle 2V\) by a thermally insulated piston. Initially both parts contain air at the temperature of \(\displaystyle T_0=300~\rm K\), and at the pressure of \(\displaystyle p_0=10^5~\rm Pa\). An electric heater of resistance \(\displaystyle R=100\,\Omega\), rated at \(\displaystyle U=230\) V, is built into that part which has a volume of \(\displaystyle V=2\) litres. The heater is operated until the values of the volume of the gas at the two parts ``swap over'', that is, in accordance with the *figure* the volume of the gas at the left hand-side of the container becomes \(\displaystyle V\), and the volume of the gas at the right hand-side becomes \(\displaystyle 2V\).

\(\displaystyle a)\) What are the temperature values at the two parts of the container at the end?

\(\displaystyle b)\) How long did the process take?

(5 points)

**P. 4829.** Two stars of different masses \(\displaystyle m_1\) and \(\displaystyle m_2\) are moving in the gravitational field of each other, while there are no other forces exerted on them. At a certain moment the distance between them is \(\displaystyle d_0\), and their velocities (magnitude and direction) are such, as if they revolve about their common centre of mass at an angular speed of \(\displaystyle \omega_0\).

\(\displaystyle a)\) What is the maximum value of \(\displaystyle \omega_0\) if \(\displaystyle d_0\) is the greatest distance between the two stars, and what is the minimum value of \(\displaystyle \omega_0\) if \(\displaystyle d_0\) is the least distance between the two stars?

\(\displaystyle b)\) What is the value of \(\displaystyle \omega\) if the gravitational field cannot keep the system together?

\(\displaystyle c)\) What is the period when gravitation keeps the system together?

(6 points)