By Brian H. Chirgwin, Charles Plumpton

ISBN-10: 0080093760

ISBN-13: 9780080093765

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**Extra resources for A Course of Mathematics for Engineers and Scientists. Theoretical Mechanics**

**Sample text**

Information provided are together sufficient t o give unique answers to t h e problems concerned. We now give examples in which it is impossible to find all t h e forces uniquely a n d which correspond t o situations of very common occurrence in practice. Consider a rigid rod AB pinned a t each end a n d carrying a weight a t t a c h e d a t a point of its length, Fig. 12. ) I t is impossible, b y t h e m e t h o d s of statics alone, t o determine uniquely t h e forces acting on this system. The conditions of equilibrium furnish three linear equations between t h e four u n k n o w n forces P , Q, B} S.

Is — l o g ^ - j . 9 The arc AC is p a r t of a catenary which is shown in Fig. 18 extended to its vertex V. The coordinate axes Ox, Oy are such t h a t M lies on Ox [see ex. (ii)] and V lies on Oy. Now A M = yA = c sec ψ A, and vertical resolution for t h e symmetrical forces on t h e particle a t C gives 2TC sin\p c — 2wlß. *. t a n ^ . = Ißc. T h e length A C of chain is c ( t a n y ^ — tany> c ) so t h a t I = AM + arc AC = c sectpA + c(t&ntpA — t a n ^ t > ) . ·. /j^ -f- t a n ψ Α = I I 3c U 3c §3:1 C O N T I N U O U S L Y D I S T R I B U T E D FORCES 53 and secy^ - tany^ = 1 /4Z ΨΑ = — + secy^ 3c — 3c \ =Ύ\ΊΪ ΊΙ)' Since c s e c y ^ — c secy; c = yA — yc = 1 /4Z 51 — , 3c \ 5/ secy,c = g : - ^ + ^ j ~ W ...

The coefficient of δ 0 gives 10 sin(0 + φ) — 7 sin(ç> — 0) + 4 sin(0 + 2ψ — φ) = 0. Since φ = ψ = π / 4 , 10 — (sin0 + cos0) \L 7 4 — (cos0 - sin0) + --τ- (sin0 + cos0) = 0 . ·. , sin0 = — ,^ , cos0 = VlO yΔ or 1 tan0 = — — , ô 3 ,^ , verifying J & that the line APF K10 is vertical. )}. 34 A C O U R S E OF M A T H E M A T I C S F Fin. 9 (i). M) \\ \\ ;ζ{ψ-Φ) \ ΥΊ* FIG. 9 (ii). §2:4 PLANE STATICS AND VIRTUAL WORK 35 With the equilibrium values for 0, y>, »/> we obtain 42\ —£i,e - W # = —^ {lO(sin0 + cos0) + 7(cos0 - sin0) - 4(sin0 + cos0)(, 4Tl = 40 W W 7ÏÔ ( 1 0 ( ~ !

### A Course of Mathematics for Engineers and Scientists. Theoretical Mechanics by Brian H. Chirgwin, Charles Plumpton

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