A2 – 3A L = 15 foot <strong>in</strong>extensible cord is tensioned by a spr<strong>in</strong>g hav<strong>in</strong>ga spr<strong>in</strong>g constant of k = 1000 pounds per foot and anunstretched length of a = 0.25 feet. A W = 20 pound weighthangs from po<strong>in</strong>t A, as shown. After some analysis, the tensionT <strong>in</strong> the cord (<strong>in</strong> pounds), the angle θ of the spr<strong>in</strong>g, and thestretch u of the spr<strong>in</strong>g were found to be related to each other bythe three equationsa = ( a + u) cosθ,W = ku s<strong>in</strong>θ,T = ku cosθ.It was determ<strong>in</strong>ed from these equations that the angle θ of thespr<strong>in</strong>g is constra<strong>in</strong>ed by the equation0 = f(θ) = a – [a + W/(ks<strong>in</strong>θ)]cosθTo f<strong>in</strong>d the angle θ, plot f(θ) over the range of θ between 0 and90 ○ and locate from the graph where f(θ) = 0. Then, f<strong>in</strong>d thetension T <strong>in</strong> the cord and the stretch u <strong>in</strong> the spr<strong>in</strong>g.Answer: θ = 30.321 ○ , T = 34.2 pounds, u = 0.475 <strong>in</strong>ches.A2 – 5A bridge is supported by cables that are suspended from acircular arch, as shown. The number of cables is 2n + 1. Thecables are tensioned to equal tensions T. After some analysis, <strong>in</strong>order to support a 100 kilo-pound weight, it has beendeterm<strong>in</strong>ed that the tension <strong>in</strong> the cables isT =1+2n∑Ws<strong>in</strong> kθ2k = 1 [cos kθ−1/(n + 1)] +s<strong>in</strong>2( kθ)πwhere θ = . Graph the tension T versus n for a range of2 ( n + 1)n between 1 and 10.Answer:T = 34.3, 23.1, 17.7, 14.4, 12.2 ,10.5, 9.27, 8.29, 7.50, 6.84 kilo - pounA2 – 4After analyz<strong>in</strong>g the triangular truss given <strong>in</strong> problem A2 – 2, itwas determ<strong>in</strong>ed that the unknown <strong>in</strong>ternal forces F AB and F ACand the unknown reactions at B and C, denoted by R Bx , R By , andR Cx , are related to each other by the five equations−FFFABAB− F− Fcosθ+ FAC= 0,s<strong>in</strong>θ= 500cosθ+ RBx= 0,s<strong>in</strong>θ+ RBy= 0,+ R = 0ABABACCx<strong>in</strong> which θ = 60 ○ . Plac<strong>in</strong>g the unknowns <strong>in</strong> the column vector x= [F AB F AC R Bx R By R Cx ] T , write these equations <strong>in</strong> matrix-vectorform as Ax = b. Then write a short program, f<strong>in</strong>d A -1 and then x= A -1 b.Answer: F AB = 577 lb, F AC = 289 lb, R Bx = -289 lb, R By = 500 lb,R Cx = 289 lb.10
A2 – 6A stone ornamental bridge be<strong>in</strong>g considered as a gateway to aEuropean city comb<strong>in</strong>es Greek and gothic elements. The cablesare not just ornamental. They also pull together beam A andpillars B. The cables are tensioned to the same tension to pulldown on the beam with a force of 2,500 pounds. The number ofcables is 2n. After some analysis, it has been determ<strong>in</strong>ed thatthe tension <strong>in</strong> the cables isT =2Hn∑250022 2k = 1 L [2 − k /( n + 1)] + H [ k /( n +kwhere L = 7 feet and H = 25 feet. Graph the tension T versus nfor a range of n between 5 and 10.Answer: T = 361, 303, 261, 229, 205, 185 pounds.1)]211