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A Course of Mathematics for Engineers and Scientists: by Brian H. Chirgwin

By Brian H. Chirgwin

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Such a force can often be determined by violating the corresponding constraint in the virtual displacement. ) The Principle of Virtual Work can be used to compare two sets of forces. If the two sets do the same work in an arbitrary virtual displace- 28 A COURSE OF MATHEMATICS ment, they are equivalent sets of forces, in the sense of § 2:2. The Principle is capable of greater generalisation than the methods of resolution and is used extensively in more advanced mechanics to discuss very general systems and obtain results of a wide applicability.

Prove that the distance 8/ (3 A B is — log -9 2 ). The arc AC is part of a catenary which is shown in Fig. 18 extended to its vertex V. The coordinate axes Ox, Oy are such that M lies on Ox [see ex. (ii)] and V lies on O y. Now AM = y A =c sec v A, and vertical resolution for the symmetrical forces on the particle at C gives 2 To sinve = 2w1/3. But T = wyc= we sect')(. •. tanve = //3c. The length AC of chain is c(tanvA — tantpc) so that 1= AM + are A C = c sec TA c(tanyA — tanzpo). •. •. •. 1 = sec2 13/ 27c 5/ 27c 3c \ 1( 4/ 2 3c + 4/ ) SeCIPC — tanzy — \2 13/ 27c 12 3c 8/ ) 23 (c\ 2 88 36 / ) -1- 272 9(c\ 4 64 ( 1 ) 3c 81 9 c2 • 0.

1). If there is a tension T in a string joining BD, the virtual work of this force in the general position shown is —T 6(B D) = T 6(2a cosT) = 2Ta sincp 699. , zero virtual work, is 3 Wa cos0 60 Wa coscp 69) 2Ta Sep = O. (3) Since, in equilibrium, BD = a, cos q) = 1, and therefore cos 0 = 2, eqn. , 60 = —6(p. , 2 T = W/1/3. Note that the tension in this special case has been calculated after writing down the equation of virtual work for the system under consideration in the general case. We do not insert the values of 0 and 97 corresponding to the specified configuration until the equation of virtual work (3) and the first order variation (1) of the equation of constraint have been written down.

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