Three pulleys rotate fixed on the same shaft, in the pulleys, there are ropes with negligible mass that support spheres. Data: pulley 1, r₁ = 0.2 m and m₁ = 2.7 kg, pulley 2, r₂ = 0.4 m, pulley 3, m₃ = 1.8 kg. Determine:
a) If m₂ = 4.0 kg, what is the radius of the pulley 3 so that the torque of the forces on the system, relative to the axis, be equal to zero;
b) If r₃ = 0.8 m, what is the mass attached to pulley 2, so the system rotates clockwise relative to the shaft.

A homogeneous rod AOB with a constant cross-section and weight 15 N is folded, making a right angle in O, AO = 1 m and BO = 0.5 m. The bar is suspended by the point O. Determine:
a) The angle α between segment AO and the vertical in the position of equilibrium;
b) The magnitude of the horizontal force that should be applied in point A, in the AOB plane, so that AO and BO have the same inclination relative to the horizontal;
c) In the case of item (b), find the magnitude of the reaction force at the point of suspension O.

A hemisphere with weight W rests on a flat horizontal plane. At the A of the diameter AB is applied a force F, the hemisphere inclines until the AB makes with the horizontal plane an angle α. Calculate, this angle knowing that the center of gravity of the hemisphere lies at a distance from the center equal to 3/8 of the radius.

Six forces of the same magnitude F act on a solid on the sides of a regular hexagon of side L. Calculate the torque of these forces relative to the axis passing through the center and perpendicular to the solid.

A crane with weight P_g, the distance between the rails in which it is supported is D. A load with weight P_l lies at a distance d from one of the rails. Determine the reaction force of the crane on the rails by lifting the load with an acceleration a=g, where g is also the acceleration due to gravity.