**P. 4573.** The coefficient of static friction between the fixed inclined plane of angle of inclination and the block of mass *M*, shown in the *figure, *is . The radii of the two alike pulleys are *r* and *R* (*R*>*r*), there is no friction at the axles, and the mass of the pulleys is negligibly small with respect to the other masses. Two pieces of rope are vertical, and one is parallel to the slope. The mass of the block at the bottom is *m*. At what *m*/*M* mass-ratio can the system be in equilibrium?

*Data:* =30^{o}, =0,2 and .

(4 points)

**P. 4574.** A right triangle shaped wedge of mass *m*=0.5 kg and of angle 30^{o} is placed to the top of a fixed slope of length =1.8 m, and of angle of elevation =30^{o}, as shown in the *figure.* The vertical height of the wedge is exactly half of the height of the slope *H*. The wedge is loaded with the small block of mass *m* as shown in the figure.

This system is assembled in two samples. In case of one of them the small block is fixed to the wedge and in the other case the block is not fixed. The two wedges are released from rest at the same time. Friction is negligible at any surfaces.

*a*) Determine the ratio of the times during which the wedges reach the bottom of the slopes.

*b*) What are the forces exerted by the small blocks on the wedges in the two cases?

(5 points)

**P. 4576.** The outer radius of a spherical shell, made of insulating material, is *r*_{2}=0.2 m, and its inner radius is *r*_{1}=0.15 m. The relative dielectric constant of its material is _{r}=8.4. The shell is charged positively, and the charge is distributed evenly such that the charge density is =10^{-5} C/m^{3}. Measured from the centre of the spherical shell where are those points at which the magnitude of the electric field strength is

*a*) *E*_{1}=3^{.}10^{3} V/m,

*b*) *E*_{2}=3^{.}10^{4} V/m?

(5 points)

**P. 4581.** A frictionlessly moveable piston of mass *m* confines *n*=0.25 moles of diatomic gas at a temperature of *T*_{1}=300 K in a vertical cylinder, closed at its bottom, of cross section *A*=2 dm^{2}. The piston is at a distance of *h*_{1}=3 dm from the bottom and the ambient pressure is *p*_{0}=10^{5} Pa.

The gas is heated in two steps. The first step lasts until the piston reaches the stoppers, in this stage the piston is *h*_{2}=5 dm from the bottom of the cylinder. The added heat in the second step, which begins when the piston reaches the stopper, is times as much as the added heat in the first step.

*a*) Determine the mass *m* of the piston.

*b*) What is the total force exerted by the stoppers on the piston at the end of the second heating step?

*c*) Plot the pressure of the gas as a function of the absolute temperature.

(5 points)