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 № Condition free/or 1.5\$ 281 A horizontally oriented uniform rod AB of mass m = 1.40 kg and length L0 = 100 cm rotates freely about a stationary vertical axis OO' passing through its end A. The point A is located at the middle of the axis OO' whose length is equal to L= 55 cm. At what angular velocity of the rod the horizontal component of the force acting on the lower end of the axis OO' is equal to zero? What is in this case the horizontal component of the force acting on the upper end of the axis? 282 The middle of a uniform rod of mass m and length 1 is rig-idly fixed to a vertical axis OO' so that the angle between the rod and the axis is equal to 0 (see Fig. 1.71). The ends of the axis OO' are provided with bearings. The system rotates without friction with an angular velocity co. Find: (a) the magnitude and direction of the rod's angular momentum M relative to the point C, as well as its angular momentum relative to the rotation axis; (b) how much the modulus of the vector M relative to the point C increases during a half-turn; (c) the moment of external forces N acting on the axle OO' in the process of rotation. 283 A top of mass m = 0.50 kg, whose axis is tilted by an angle θ = 30° to the vertical, precesses due to gravity. The moment of inertia of the top relative to its symmetry axis is equal to I = 2.0 g*m2, the angular velocity of rotation about that axis is equal to ω = 350 rad/s, the distance from the point of rest to the centre of inertia of the top is l = 10 cm. Find: (a) the angular velocity of the top's precession; (b) the magnitude and direction of the horizontal component of the reaction force acting on the top at the point of rest. 284 A gyroscope, a uniform disc of radius R = 5.0 cm at the end of a rod of length L= 10 cm (Fig. 1.73), is mounted on the floor of an elevator car going up with a constant acceleration w = 2.0 m/s^2. The other end of the rod is hinged at the point 0. The gyroscope precesses with an angular velocity n = 0.5 rps. Neglecting the friction and the mass of the rod, find the proper angular velocity of the disc. 285 A top of mass m = 1.0 kg and moment of inertia relative to its own axis I = 4.0 g•m2 spins with an angular velocity w = 310 rad/s. Its point of rest is located on a block which is shifted in a horizontal direction with a constant acceleration w = 1.0 m/s^2. The distance between the point of rest and the centre of inertia of the top equals L = 10 cm. Find the magnitude and direction of the angular velocity of precession w'. 286 A uniform sphere of mass m = 5.0 kg and radius R = 6.0 cm rotates with an angular velocity ω = 1250 rad/s about a horizontal axle passing through its centre and fixed on the mounting base by means of bearings. The distance between the bearings equals l = 15 cm. The base is set in rotation about a vertical axis with an angular velocity ω' = 5.0 rad/s. Find the modulus and direction of the gyroscopic forces. 287 A cylindrical disc of a gyroscope of mass m = 15 kg and radius r = 5.0 cm spins with an angular velocity w = 330 rad/s. The distance between the bearings in which the axle of the disc is mounted is equal to L = 15 cm. The axle is forced to oscillate about a horizontal axis with a period T = 1.0 s and amplitude fm = 20°. Find the maximum value of the gyroscopic forces exerted by the axle on the bearings. 288 A ship moves with velocity v = 36 km per hour along an arc of a circle of radius R = 200 m. Find the moment of the gyroscopic forces exerted on the bearings by the shaft with a flywheel whose moment of inertia relative to the rotation axis equals I = 3.8*103 kg*m2 and whose rotation velocity n = 300 rpm. The rotation axis is oriented along the length of the ship. 289 A locomotive is propelled by a turbine whose axle is parallel to the axes of wheels. The turbine's rotation direction coincides with that of wheels. The moment of inertia of the turbine rotor relative to its own axis is equal to I = 240 kg•m^2 . Find the additional force exerted by the gyroscopic forces on the rails when the locomotive moves along a circle of radius R = 250 m with velocity v = 50 km per hour. The gauge is equal to L = 1.5 m. The angular velocity of the turbine equals n = 1500 rpm 290 What pressure has to be applied to the ends of a steel cylinder to keep its length constant on raising its temperature by 100 °C? 291 What internal pressure (in the absence of an external pressure) can be sustained (a) by a glass tube; (b) by a glass spherical flask, if in both cases the wall thickness is equal to Δr = 1.0 mm and the radius of the tube and the flask equals r = 25 mm? 292 A horizontally oriented copper rod of length L = 1.0 m is rotated about a vertical axis passing through its middle. What is the number of rps at which this rod ruptures? 293 A ring of radius r = 25 cm made of lead wire is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring. What is the number of rps at which the ring ruptures? 294 A steel wire of diameter d = 1.0 mm is stretched horizontally between two clamps located at the distance l = 2.0 m from each other. A weight of mass m = 0.25 kg is suspended from the midpoint O of the wire. What will the resulting descent of the point O be in centimetres? 295 A uniform elastic plank moves over a smooth horizontal plane due to a constant force F0 distributed uniformly over the end face. The surface of the end face is equal to S, and Young's modulus of the material to E. Find the compressive strain of the plank in the direction of the acting force. 296 A thin uniform copper rod of length l and mass m rotates uniformly with an angular velocity ω in a horizontal plane about a vertical axis passing through one of its ends. Determine the tension in the rod as a function of the distance r from the rotation axis. Find the elongation of the rod. 297 A solid copper cylinder of length L = 65 cm is placed on a horizontal surface and subjected to a vertical compressive force F = 1000 N directed downward and distributed uniformly over the end face. What will be the resulting change of the volume of the cylinder in cubic millimetres? 298 A copper rod of length L is suspended from the ceiling by one of its ends. Find: (a) the elongation dl of the rod due to its own weight; (b) the relative increment of its volume dV/V. 299 A bar made of material whose Young's modulus is equal to E and Poisson's ratio to n, is subjected to the hydrostatic pressure p. Find: (a) the fractional decrement of its volume; (b) the relationship between the compressibility b and the elastic constants E and n. Show that Poisson's ratio n cannot exceed 1/2. 300 One end of a steel rectangular girder is embedded into a wall (Fig. 1.74). Due to gravity it sags slightly. Find the radius of curvature of the neutral layer (see the dotted line in the figure) in the vicinity of the point O if the length of the protruding section of the girder is equal to L = 6.0 m and the thickness of the girder equals h= 10 cm. 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