**P. 4498.** A plano-convex and a plano-concave lens exactly fit to each other along their curved surfaces. The refractive index of the material they are made of is 1.5, and the focal length of the plano-convex lens is 40 cm.

*a*) What is the power of each lens?

*b*) What is the radius of the curved surfaces of the lenses?

(3 points)

**P. 4499.** The mass of an empty balloon is measured with a sensible electronic scale. The mass is 3.7 g. After blowing the balloon the reading on the scale was 4.5 g. The ,,perimeter'' of the balloon, whose shape could well be approximated as a sphere, was measured as 73 cm.

*a*) How much air was blown into the balloon?

*b*) What is the excess pressure inside the balloon, if the temperature in the room is 25 ^{o}C?

*c*) What is the drag force exerted on the balloon, when it descends uniformly?

(4 points)

**P. 4500.** A small bead can slide from *A* to *B* along the paths shown *in the figure. *The paths are circular arcs lying in a vertical plane, and they are symmetrical about the straight line through the points *A* and *B*. The angle between the straight line and the horizontal is 45^{o}.

*a*) Along which path will the bead reach point *B* in a shorter time, and what can we state about the final speed in each case, if there is no friction?

*b*) What can be stated about the final speeds if friction is not negligible, and the coefficient of friction is the same in both paths?

(4 points)

**P. 4501.** A fixed U shaped small narrow tube is turned upside-down and a thread of length is hanging in it as shown *in the figure.* Initially the length of the thread is /2 in each side. This is an unstable equilibrium since friction is negligibly small.

*a*) If the thread begins to slide to one side what will its speed be when it is straight?

*b*) What will the speed of the thread be when its momentary centre of mass is just at the same level as that end of the thread which is moving up?

(4 points)

**P. 4502.** A disc is given a push and slides on the top a horizontal table. It stops in a distance of *L*_{0}. When the disc is placed to the rim of the table, which is at a height of *H*, and is given the same initial speed it reaches the ground at a distance of *d* from the table.

*a*) What is the kinetic frictional coefficient between the table and the disc?

*b*) At what distance from the table will the disc reach the ground if initially it is not exactly at the rim of the table but at a distance of *L* from the rim, (*L*<*L*_{0}), the disc is given the same initial speed, and slides towards the rim of the table.

(*Data:* *H*=*L*_{0}=*d*=0.8 m; *L*=0.6 m.)

(4 points)

**P. 4503.** The pressure and the volume of a sample of 0.4 moles Helium gas is doubled. Initially its pressure was 10^{5} Pa, and its volume was 10 dm^{3}. Determine those quantities from the list below which can be given uniquely:

*a*) The work done on the gas;

*b*) The heat given to the gas;

*c*) The change in the energy of the gas.

(4 points)

**P. 4505.** The values the resistance of the resistors connected to the voltage supply of *U*=120 V, as shown *in the figure* are unknown. When the switch is closed the reading on the voltmeter changes by 16 V, and the power dissipated by the resistor *R*_{1} changes by 51.2 W.

*a*) Using the condition given to the change in the voltage determine the ratio of the resistances: *R*_{1}/*R*_{2}.

*b*) What are the values of the resistance of the resistors *R*_{1} and *R*_{2}?

(4 points)