Problem S1–2008 (10 points) 


The inverse mechanism is located in a horizontal
plane. Sliders A and B can
move along an axis which are taking place through point O. It is need to
prove, that in the position represented in figure (AD ^ OC, and corners, which rods AD and BC make with axis OB,
are identical) the mechanism will be in equilibrium at condition _{}. 
Problem S2–2008 (8 points) 


The uniform rod 1
with one of its ends is fixed with disk 2 and the other end is hinged at B with a disk 3.
Disk’s radiuses are identical and equal r, and their weights are
neglecting. Disks rest on the surface having the form of a semicircle with
radius R = 5r. Coefficient of sliding friction between disks and a
surface is f. Coefficient of rolling friction between disk 3 and the
same surface is δ. Rod
length is _{}. Determine maximum value of the angle j between a rod 1 and a horizontal
plane when the rod is in equilibrium. 
Problem K1–2008 (10 points) 


In the planar mechanism crank OA rotates with constant angular
velocity ω. OA = BC = = CD = CE =DF = l, _{}. Determine velocity and acceleration of point B for mechanism position when _{}, _{}, a = β = 30°. 
Problem K2–2008 (10 points) 


The square plate has uniformly accelerated rotation around
a vertical axis. Its initial angular velocity is distinct from zero. Point M moves on diagonal AB of a plate so, that the absolute acceleration vector all the movement time lays in a plate’s plane.
During the initial time point M is in position A. Prove that in a
considered case point M will never be lowered below 
Problem D1–2008 (7 points) 

Maximum
horizontal range of a stone thrown from the immovable catapult is s. Find the maximum possible horizontal range of the stone thrown from the very catapult but fixed on the platform the mass of which together with the catapult is n times heavier than the stone mass. The platform is on the horizontal plane. At the start time it is motionless. Any resistance can be neglected.^{} 

Problem D2–2008 (7 points) 


The uniform Lshaped rod with equal crosssections AB = l and BC = 2l is located in a vertical plane. 
Problem D3–2008 (8 points) 


Two identical homogeneous smooth cylinders with
radius R are leant against a vertical wall. The bottom cylinder displaced to the right on a
horizontal plane. The top cylinder began to fall on a vertical. The system has come to
motion. Find final
velocity of the bottom cylinder. 
Problem D4–2008 (10 points) 


The pulley 1 of
weight m_{1} and radius r_{1} is set in
rotation by constant torque of moment M. From pulley 1 through a
deformable cable motion is transferred to the pulley 2 of weight m_{2}
and radius r_{2}. In this case a cable it is possible replace with
a spring which force of a tension is directly proportional to deformation of
a rectilinear part of a cable, the proportionality coefficient is equal c. Determine the law
of angular velocity of the pulley 1. Initially the system was in equilibrium. Both pulleys are assumed to be homogeneous
cylinders. Neglect deformations of curvilinear parts of a cable and a
friction. 