**G. 598.** A piece of ice having temperature \(\displaystyle 0\,{}^\circ\)C is thrown into a thermally insulated container, filled with water of volume \(\displaystyle 5~{\rm dm}^3\) at a temperature of \(\displaystyle 50\,{}^\circ\)C. What is the final amount of water in the container if after the ice melts the temperature of the system is \(\displaystyle 15\,{}^\circ\)C?

(3 points)

This problem is for grade 1 - 9 students only.

**P. 4918.** A plane approaches the runway at a speed of \(\displaystyle v\). The depression angle of the gliding is \(\displaystyle \alpha\). When it is at a height of \(\displaystyle H\) above the ground, it leaves the straight path and follows a circular trajectory along which its speed remains \(\displaystyle v\). When it reaches the runway it just flies horizontally.

\(\displaystyle a)\) What is the radius of the circular trajectory?

\(\displaystyle b)\) For how long does the plane fly along the circular arc?

\(\displaystyle c)\) At most what is the percent increase of the weight of the pilot?

*Data:* \(\displaystyle v=70~{\rm m}/{\rm s}\), \(\displaystyle \alpha=3^\circ\), \(\displaystyle H=100\) m.

(4 points)

**P. 4919.** A trolley of mass \(\displaystyle m_1\) was placed to a horizontal, rigid pair of rails at some height as shown in the *figure.* There is a small ball of mass \(\displaystyle m_2\) attached to one end of a thread, whose other end is attached to the bottom of the trolley. (The thread hangs between the rails. The length of the thread is much greater than the size of the ball and the trolley.)

If the ball is displaced a bit from its stable equilibrium position in the direction perpendicular to the rails, then it swings with a period of \(\displaystyle T_1\). If the ball is displaced a bit parallel to the rails, while the trolley is fixed, and then both the ball and the trolley are released, then the period of the oscillation is \(\displaystyle T_2\). What is the ratio of \(\displaystyle m_2/m_1\), if \(\displaystyle T_1/T_2=2\)? (Friction and air drag are negligible.)

(5 points)

**P. 4920.** A police car is approaching a stationary observer at a constant speed, and then it is moving away from it at the same constant speed.

\(\displaystyle a)\) What is the speed of the police car, if the observer detects that the sound of the siren of the car when it is approaching is higher by a minor third than the detected sound when it is moving away?

\(\displaystyle b)\) What is the percent decrease in the difference between the detected sound of the siren of the approaching and leaving police car for an observer, who is in another car travelling towards the police car from the front at a speed of 36 km/h?

The speed of sound is 330 m/s.

(4 points)

**P. 4921.** Frictionless movable pistons are in a horizontal cylinder, closed at both ends, and confine samples of diatomic, ideal gas of the same type and of the same volume of \(\displaystyle V_0=2~\rm dm^3\) at a pressure of \(\displaystyle p_0=10^4\) Pa, and at the same temperature. The pistons are connected by a spring the length of which when it is not compressed or stretched is the same as the length of the cylinder. Between the pistons there is vacuum.

\(\displaystyle a)\) What is the energy stored in the spring in its compressed position described above?

\(\displaystyle b)\) The gas samples are heated slowly in the same way, and their absolute temperature is tripled. What will the pressure of the gas samples be at the end of the heating process?

\(\displaystyle c)\) How much heat was added to the system, if the losses are neglected?

(5 points)

**P. 4923.** An insulated metal disc of radius 0.250 m is rotating with 1000 revolutions per minute. Determine the potential difference between the rim of the disc and its centre if

\(\displaystyle a)\) there is no external magnetic field;

\(\displaystyle b)\) it is in uniform magnetic field of magnitude 10.0 mT, which is perpendicular to the plane of the disc.

(5 points)