**P. 4154.** Andrew and Ben are running uniformly along a circular track. If they run in opposite directions between two meetings Andrew steps 225 and Ben steps 220, while if they run in the same directions they have to step ten times as much. Ben is quicker then Andrew.

*a*) How many steps is one lap for Andrew and for Ben?

*b*) Determine the ratio of the speeds of the two runners.

(4 points)

**P. 4158.** A uniform solid disc of radius 10 cm and of mass 5 kg is rotated at an angular speed of 120 1/s along its horizontal symmetry axis, and carefully, without giving any initial speed to it, it is placed to a horizontal surface. The coefficient of friction between the disc and the surface is 0.4.

Determine the work done by the frictional force on the disc, while it covers a distance of 5 m:

*a*) with using the definition of work done;

*b*) with using the work-energy theorem.

(Neglect air resistance and rolling friction, and calculate with *g*=10 m/s^{2}.)

(5 points)

**P. 4159.** Two loud speakers, in the open air, emit the same frequency of sound in phase. One of them is at the origin of the coordinate system, and the other is at the point *x*_{1}=0, *y*_{1}=2.4 m. An observer is standing at the point *x*_{2}=40 m, *y*_{2}=0, and then along a straight line parallel to the *y* axis he walks to the direction of positive and negative *y*. While he is walking he finds that the intensity of sound decreases to both directions. What can the frequency of sound be? Along the same line can he find similar points?

(4 points)

**P. 4161.** The capacitors arranged as shown in the *figure* are all neutral. At a certain instant switch *K*_{1} is closed.

*a*) Find the voltage across each capacitors after switch *K*_{1} is closed.

*b*) After a short time switch *K*_{2} is closed as well. Determine the energy of the condensers.

*c*) How much heat was generated in the system after closing switch *K*_{2}?

(5 points)

**P. 4162.** A magnetized needle can swing horizontally inside a solenoid whose symmetry axis is horizontal. The length of the solenoid is 25 cm, and the number of its turns is 100. When there is no current in the solenoid the needle makes 10 total swings in each minute due to the magnetic field of the Earth. Now the solenoid is placed such, that its symmetry axis is perpendicular to the magnetic meridian. The needle makes 51 swings in each minute when the current in the solenoid is 1 A.

*a*) What is the horizontal component of the magnetic field of the Earth at the location of the experiment?

*b*) How many swings does the needle make if the direction of the current in the solenoid is changed? (Its magnitude remains the same.)

(4 points)

**P. 4163.** Use a converging lens of diameter *d* and of power *D* dioptre and project the image of the Sun onto a white screen.

*a*) What is the size of the Sun on the screen?

*b*) By what factor will the image of the Sun on the screen be brighter than the rest of the screen which is directly shone by the Sun?

The angle subtended by the Sun is 32 arcminutes.

*Data: D*_{1}=12 m^{-1}, *d*_{1}=35 mm; and , *d*_{2}=60 cm.

(4 points)