**P. 4340.** A small ball of mass *m*=0.4 kg is hanging on a thread of length =1 m, and touches another ball of mass *m* and of the same radius. This second ball is on a horizontal plane and is attached to one end of a horizontal spring of spring constant *D*=10 N/m, as shown in the *figure.* Initially the spring is unstretched and its other end is fixed. The simple pendulum is deflected at an angle of =20^{o}, and it is released without any initial speed.

*a*) At what speed do the two balls collide? What is the percentage error that is made if the speed of the ball is calculated from the formula of the period of the pendulum.

*b*) How much time elapses until the pendulum gets back again to the position where its displacement is the greatest, having collided with the other ball totally elastically? (Friction is negligible.)

(5 points)

**P. 4342.** There are three vases on the table, none of them is taller than 40 cm. The hole in all of them is a vertical cylinder. The first vase can be filled with 2 dl water, the second one with 3 dl, and the third with more than 6 dl water. The area of the cross section of the first vase is 10 cm^{2}, the area of the cross section of the other two vases is 20 cm^{2}. There is the same amount of water in each vase. It can be observed that if a specific tuning fork is placed just above the vases, its sound is getting louder. The width of the bottom of each vase is 1 cm, the speed of sound in air is 340 m/s, and the frequency of the tuning fork is less than 1 kHz.

*a*) What is the amount of water in each vase?

*b*) What is the frequency of the tuning fork?

*c*) What is the height of the third vase?

(5 points)

**P. 4343.** The three cylinders shown in the *figure* have the same cross section. In the cylinders there is the same type of ideal gas, and all the cylinders are closed with easily movable pistons of the same mass. The pistons are rigidly attached to each other. Initially the three samples of gas have the same density, the same temperature of *t*=27 ^{o}C, and the same height of =20 cm.

How much do the pistons rise if the temperature of the gas in the middle cylinder is increased to *t*'=117 ^{o}C, while the temperature of the gas in the two bordering cylinders is kept constant?

(4 points)

**P. 4346.** A hundred years ago in 1911 Rutherford wrote in his article, that the speed of an particle, which is moving towards the centre of a gold atom, decreases to zero at that distance, measured from the centre of the gold atom, where its electric potential energy is equal to its initial kinetic energy. The electric potential energy is partly due to the positive charge at the centre of the atom, and partly due to the negative charge which is uniformly distributed in the atom. The radius of the gold nucleus is 10^{-10} m.

What is this particular distance, if the speed of the particle is 2^{.}10^{7} m/s?

(5 points)