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- A bar ABC revolves in a horizontal plane about a vertical axis at the midpoint C (see figure). The bar, which has a length 2L and crass-sectional area A, revolves at constant angular speed at. Each half of the bar (AC and BC) has a weight W, and supports a weight W2at its end. Derive the following formula for the elongation of one-half of the bar (that is. the elongation of either AC ar BC). =L223gEA(w1+3w2) in which E is t he modulus of elasticity of the material of the bar and g is the acceleration of gravity.A round bar ABC of length 2L (see figure) rotates about an axis through the midpoint C with constant angular speed w (radians per second). The material of the bar has weight density y. (a) Derive a formula for the tensile stress a’ in the bar as a function of the distance x from the midpoint C. (b) What is the maximum tensile stress a max?The L-shaped arm ABCD shown in the figure lies in a vertical plane and pivots about a horizontal pin at A. The arm has a constant cross-sectional area and total weight W. A vertical spring of stiffness k supports the arm at point B. (a) Obtain a formula for the elongation of the spring due to the weight of the arm. (b) Repeat part (a) if the pin support at A is moved to D.
- Solve the preceding problem if the collar has mass M = 80 kg, the height h = 0.5 m, the length L = 3.0 m, the cross-sectional area A = 350mm2. and the modulus of elasticity E = 170 GPa.A uniform bar AB of weight W = 25 N is supported by two springs, as shown in the figure. The spring on the left has a stiffness k[= 300 N/m and natural length Lt=250 mm. The corresponding quantities for the spring on the right are k2= 400 N/m and L^ = 200 mm. The distance between the springs is L = 350 mm, and the spring on the right is suspended from a support that is a distance it = SO mm below the point of support for the spring on the left. Neglect the weight of the springs. (a) At what distance x from the left-hand spring (figure part a) should a load P = 18 N be placed in order to bring the bar to a horizontal position? (b) If P is now removed, what new value of k{is required so that the bar (figure part a) will hang in a horizontal position underweight If? (c) If P is removed and kt= 300 N/m. what distance b should spring ktbe moved to the right so that the bar (figure part a) will hang in a horizontal position under weight II"? (d) If the spring on the left is now replaced by two springs in series (kt= 300 N/m, kt) with overall natural length Lt= 250 mm (see figure part b). what value of k; is required so that the bar will hang in a horizontal position under weight IF?Find support reactions at 4 and Band then use the method of joints to find all member forces. Let b = 3 m and P = 80 kN.
- A long, slender bar in the shape of a right circular cone with length L and base diameter d hangs vertically under the action of its own weight (see figure). The weight of the cone is W and the modulus of elasticity of the material is E. Derive a formula for the increase S in the length of the bar due to its own weight. (Assume that the angle of taper of the cone is small.)A space truss is restrained at joints A, B. and C, as shown in the figure. Load P acts in the +z direction at joint Band in the -z directions at joint C. Coordinates of all joints are given in terms of dimension variable L (see figure). Let P = 5 kN and L = 2 m. (a) Find the reaction force components Az and Bx. (b) Find the axial force in truss member AB..17 A mountain-bike rider going uphill applies torque T = Fd(F = l5lb, d = 4 in.) to the end of the handlebars ABCD by pulling on the handlebar extenders DE. Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at A). Segments AB and CD are prismatic with lengths L, = 2 in.andL3 = 8.5 in, and with outer diameters and thicknesses d01 = 1.25 in. 101 = 0.125 in. and d03 = O.87in.,i03 = 0.ll5in, respectively as shown. Segment BC’ of length L, = 1.2 in. however. is tapered, and outer diameter and thickness vary linearly between dimensions at B and C. Consider torsion effects only. Assume G = 4000 ksi is constant. Derive an integral expression for the angle of twist of half of the handlebar tube when it is subjected to torque T = Fd acting at the end. Evaluate ‘b1-, for the given numerical1ues.
- Solve the preceding problem for a steel pipe column (E = 210 GPa) with length L = 1.2 m, inner diameter d2= 36 mm, and outer diameter d2=40 mm.Solve the preceding problem for sx= 11 MPa and ??y= -20 MPa (see figure).Solve the preceding problem for a W 250 × 89 steel column having a length L = 10 m. Let E = 200 GPa.