**P. 4046.** The *figure* shows a smooth vertical path. On the left-handside there is a ramp, the incline angle is =45^{o}, than it is continued with a semi-circle of radius *R*, and then there is a parabola at the end. The symmetry axis of the parabola is vertical, and the parabola is open above. If a body travel through the path without friction, the normal force exerted on it does not change abruptly at the border of the semicircle and the parabola.

*a*) What is the height *h* of the parabola?

*b*) What is the least height *H*, measured from the ground, at which an object should be released in order to go through the whole path?

(5 points)

**P. 4048.** Inside a spherical container of mass *M*=385 kg and of cross-section *A*=1.6 m^{2} there is a hanging weight, which can slide easily, above a spring, winded round a thin stick. The mass of the weight is *m*=1.5 kg, and the spring constant is *D*=155 N/m. (At the beginning the spring is unstretched.) The container is falling from a weather balloon, which is at high altitude, without initial speed. At the beginning of the fall the hanging of the weight is ceased. At what distance from the bottom of the weight should the electric sensor be placed in order to switch on the device which opens the parachute when the container reaches a speed of *v*=80 m/s? Assume that the density of air is constant.

(5 points)

**P. 4049.** A sample of diatomic gas of certain mass is taken from state *A* to state *B* as shown in the *figure;* *p*_{0}=10^{5} Pa, *V*_{0}=2 dm^{3}.

*a*) Determine the pressure and the volume of the gas at the intermediate state *C* until which only *Q*_{AC}=3150 J heat was added to the gas.

*b*) By what factor does the internal energy of the gas changed during process *A**C*?

*c*) How much work was done by the expanding gas during process *A**C*?

(4 points)