国际工程力学历届竞赛真题eng_2015-br

Brain-ring 2015

Statics 1 Find out the module of the three forces resultant R: F1=3i?2jN;F2=6jN;F3=5i+8kN. A 2 A cylinder of G = 90 кН and radius R = 0,55 m is covered by rope loop, attached to a fixed point in the wall. Find the value of the dimen-sion а, determining the position of the vanishing point of the rope from the cylinder, if the tension forces of loop branches are Т1 = Т2 = 75 kN. A cube of 200 N weight is held in equilibrium by two smooth surfaces. Determine the distance AB from the angle A vertex to the point of appli-cation of the the vertical plane reaction, if the cube edge length is63 cm. The top end of the homogeneous rod AB of mass m is based on a rough vertical wall. A cable BC is connected with lower end of rod AB (at point B). The rod makes an angle β with the vertical. Determine the minimal value for the angle β for the case of rod equilibrium if АС = 1,5l, СВ = 4l, friction coefficient f = 0,2. Determine the value of the resultant reaction of rods 1 and 2 acting on the node С, if F = 40 Н. Determine the distance from the center of gravity of the plane fig-ure to the diagonal АВ (for reference: the center of gravity of the circular sector is at a dis-B60R303 4 z45° 15 xFC30°2yA6 702424tance 2Rsinα from the center of the arc with a central angle 2α). 3αFind out the value of angle α for the equilibrium of the shown con-7 struction if Q = 0,6P, q = Q3R, a = 2R= 6r, b = 3R, с = 4,5r. 8 The shown construction is in the equilibrium under the force F loading. Specify the numbers of rods that can be replaced by the cable. AQ60оB In the shown mechanism ОА = ОВ = l, ВС = 2l. Two forces Q and F are applied to the mechanism. Determine the relation Qfor the case 9 O60оFF45оC of equilibrium of the system. 10 Determine the reaction of the hinge С, if F = 10 kN; P = 5 kN; q = 4 kN/m; M = 20 kN m. Kinematics Find out the velocity of point М for time t = 0 if this point is located on the rod АВ. It is known that s=3sin2t m, AB = 5 m, ВМ = 1 m. Determine the ρ(t) dependence of the trajectory curvature radius on time, if the coordinates of the 12 πtpoint are adjusted according to the following laws:x(t)=3cos ?7 m;y(t)=5+4sin2 (0.25πt) m. 2Determine the duration of the photographic shutter, if during the ball falling along the vertical 13 centimeter scale without initial velocity it was obtained strip on the negative lying from 25 to 28 cm scale interval? Rim points arc coordinate for the disk of 0.3m radius varies by law s(t) = 0,6t2 m. Determine the 14 angle between the velocity vector and acceleration vector of disk rim point at 0.5 seconds after the be-ginning of movement. 15 The shaft started to rotate with constant acceleration from the rest and made 3600 revolutions per 0.2 hours. Determine the angular acceleration of the shaft. Find the value of the slide О velocity at time t = 1 sec, if it is known that OМ = 4 m, x0 = 0 m, the absolute velocity of a point M is πttwice more than the velocity of the slide O, ?=π2sinрад 411 16 3B A1O217 Determine velocity of point В (АВ is vertical for the shown posi-tion) located on the rim of wheel 3, if the angular velocity of the wheel 1 ω1 = ω, ОА = 4r, АВ = 2r, ω2 = 2ω1. Rod 2 makes an angle of 30° with horizontal line. Determine the absolute velocity of point M for the shown position of the mechanism, if ω1 = 2 rad/sec, h = 10 cm. 18 A point moves along a circle in the plane of the radius R = 10 cm according to the equation 19 s(t) = 12πt–2πt2 cm. Determine the total acceleration of the point for the moment of time when the arc coordinate is equal to circle half. Find the value of point М acceleration at t = 0,5 sec, if the point is based on the rim of wheel 5, the law of wheel 1 rotating 1?1=2ln rad, and r1 = r3 = r5 =0,2r2 = 0,5r4 = 0,8 m. t?1 20 Dynamics A hammer of mass m = 0,6 kg had velocity v = 4 m/sec at hitting the nail head. The nail is under 21 the resistance force F at its movement: F = a + bx, where x – nail displacement, a = 10 N, b = 2 N/m. Determine how many hits should be applied to move the nail at a distance L = 0,2 m. АС22 lАВВ Determine the distance l of point M, separated from the bicycle wheel of radius R at position A. The movement of the point is in the ver-tical plane. Wheel center of mass velocity was equal to 5 m/s at the ini-tial position. The wheel rolled without slipping. A ball moves in a smooth curved tube and at point A it has veloci-ty vA = 0,5 m/sec. Determine the ball velocity at point C, if on the R23 curved interval of the tube the ball is under the variable driving force С F = 2mgs, R = 0,2 m. A skier of mass 75 kg acquires an initial velocity of 1 m/s due to a push and slides down the slope, making an angle of 30 ° with the horizon. Resistance force due to the motion of the skier is 24 proportional to his velocity. Determine the velocity of the skier 5 seconds after the beginning of the movement, if the value of the resistance force for the velocity of 0.5 m/sec is equal to 10 N. 1В225 АTwo homogeneous cylinders are connected by homogeneous rod АВ of 2m mass. Each of the cylinders have radius r and mass 4m and can roll along the horizontal surface without slipping. Determine the kinetic energy of the system for the shown position if the angular ve- locity of the cylinder 1 is known and it is equal to ω. A wheel of mass m rolls on a rough surface under the action of gravity and active force F = 4mg. Wheel radii are equal to R and r accordingly and R = 1,5r. The gyration radius of the wheel is iC =1.2r. Rolling resistance coefficient is δ, coefficient of static friction is f. Determine the acceleration of the wheel center of mass, if α = β = 60°. Homogeneous thin plate 1 of mass m1 = m rotates about a fixed vertical axis z with angular velocity ω0. Ball 2 of mass m2 = 0,5m is located in the groove on the plate and kept at a distance a. The ball starts to move along the gutters and compresses the spring by an amount λ. Determine the angular velocity of the plate at this time, if the distance l is known. 26 27 28 The mechanical system consists of two bodies with masses m1 = 2m and m2 = 3m. Wheel 2 is loaded by driving force Р=6mg, friction coefficient between contacting surfaces is equal to f. De-termine the acceleration of body 1, if R = 2,5r; i2х = r2. A homogeneous rod 1 of mass m located in a vertical plane and it is pivotally connected with the rod 2 of mass 3m. The rod 2 can move along the horizontal line AB. The rod 1 falls down from the protrusion N on the horizontal rod 2, rod 2 under this movements is shifted by distance s. Determine the length of the rod 1, neglect-ing friction. 29 30 & + bx&+100x = 0. Determine the resistance The dynamic equation of material point motion is &xcoefficient b values of elastic medium for the case of movement with no oscillations.