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301The bending of an elastic rod is described by the elastic curve passing through centres of gravity of rod's cross-sections. At small bendings the equation of this curve takes the form N(x)=EI(d^2y/dx^2), where N (x) is the bending moment of the elastic forces in the crosssection corresponding to the x coordinate, E is Young's modulus, I is the moment of inertia of the cross-section relative to the axis passing through the neutral layer (I = .., Fig. 1.75). Suppose one end of a steel rod of a square cross-section with side a is embedded into a wall, the protruding section being of length L (Fig. 1.76). Assuming the mass of the rod to be negligible, find the shape of the elastic curve and the deflection of the rod X, if its end A experiences (a) the bending moment of the couple N0; (b) a force F oriented along the y axis.
302A steel girder of length L rests freely on two supports (Fig. 1.77). The moment of inertia of its cross-section is equal to I (see the foregoing problem). Neglecting the mass of the girder and assuming the sagging to he slight, find the deflection X due to the force F applied to the middle of the girder.
303The thickness of a rectangular steel girder equals h. Using the equation of Problem 1.301, find the deflection X caused by the weight of the girder in two cases: (a) one end of the girder is embedded into a wall with the length of the protruding section being equal to 1 (Fig. 1.78a); (b) the girder of length 21 rests freely on two supports (Fig. 1.78b).
304A steel plate of thickness h has the shape of a square whose side equals L, with h << L. The plate is rigidly fixed to a vertical axle OO which is rotated with a constant angular acceleration b (Fig. 1.79). Find the deflection k, assuming the sagging to be small.
305Determine the relationship between the torque N and the torsion angle f for (a) the tube whose wall thickness dr is considerably less than the tube radius; (b) for the solid rod of circular cross-section. Their length L, radius r, and shear modulus G are supposed to be known.
306Calculate the torque N twisting a steel tube of length L = 3.0 m through an angle cp = 2.0° about its axis, if the inside and outside diameters of the tube are equal to d1 = 30 mm and d2 = 50 mm.
307Find the maximum power which can be transmitted by means of a steel shaft rotating about its axis with an angular velocity w = 120 rad/s, if its length L = 200 cm, radius r = 1.50 cm, and the permissible torsion angle f = 2.5°.
308A uniform ring of mass m, with the outside radius r2, is fitted tightly on a shaft of radius r1. The shaft is rotated about its axis with a constant angular acceleration b. Find the moment of elastic forces in the ring as a function of the distance r from the rotation axis.
309Find the elastic deformation energy of a steel rod of mass m = 3.1 kg stretched to a tensile strain E = 1.0·10^-3.
310A steel cylindrical rod of length L and radius r is suspended by its end from the ceiling. (a) Find the elastic deformation energy U of the rod. (b) Define U in terms of tensile strain dL/L of the rod.
311What work has to be performed to make a hoop out of a steel band of length L = 2.0 m, width h = 6.0 cm, and thickness 6 = 2.0 mm? The process is assumed to proceed within the elasticity range of the material.
312Find the elastic deformation energy of a steel rod whose one end is fixed and the other is twisted through an angle f = 6.0°. The length of the rod is equal to L = 1.0 m, and the radius to r = 10 mm.
313Find how the volume density of the elastic deformation energy is distributed in a steel rod depending on the distance r from its axis. The length of the rod is equal to l, the torsion angle to φ.
314Find the volume density of the elastic deformation energy in fresh water at the depth of h = 1000 m.
315Ideal fluid flows along a flat tube of constant cross-section, located in a horizontal plane and bent as shown in Fig. 1.80 (top view). The flow is steady. Are the pressures and velocities of the fluid equal at points 1 and 2? What is the shape of the streamlines?
316Two manometric tubes are mounted on a horizontal pipe of varying cross-section at the sections S1 and S2 (Fig. 1.81). Find the volume of water flowing across the pipe's section per unit time if the difference in water columns is equal to dh.
317A Pitot tube (Fig. 1.82) is mounted along the axis of a gas pipeline whose cross-sectional area is equal to S. Assuming the viscosity to be negligible, find the volume of gas flowing across the section of the pipe per unit time, if the difference in the liquid columns is equal to dh, and the densities of the liquid and the gas are r0 and r respectively;
318A wide vessel with a small hole in the bottom is filled with water and kerosene. Neglecting the viscosity, find the velocity of the water flow, if the thickness of the water layer is equal to h1 = 30 cm and that of the kerosene layer to h2 = 20 cm.
319A wide cylindrical vessel 50 cm in height is filled with water and rests on a table. Assuming the viscosity to be negligible, find at what height from the bottom of the vessel a small hole should be perforated for the water jet coming out of it to hit the surface of the table at the maximum distance lmax from the vessel. Find lmax.
320A bent tube is lowered into a water stream as shown in Fig. 1.83. The velocity of the stream relative to the tube is equal to v = 2.5 m/s. The closed upper end of the tube located at the height h0 = 12 cm has a small orifice. To what height h will the water jet spurt?
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