By Brian H. Chirgwin
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Additional info for A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics
If the framework rests in a vertical plane on two supports at A and B, so that AB is horizontal and below DE, prove that the tensions in the rods BF and DF are each of magnitude W and the thrust in BD is 2 W. 7. A rhombus ABCD consists of four equal uniform rods, of length a and weight W, freely jointed together at their ends. The corners A and C, B and D are joined by two light elastic strings under tension, of equal natural length and each having a modulus of elasticity equal to 2 W. The system is freely suspended from A so as to hang in a vertical plane.
2) § 2:4 PLANE STATICS AND VIRTUAL WORK 29 The detailed reasons for writing down these expressions are as follows. Equation (1) is the form which the scalar product F • 6s takes in this case: F is here the weight which acts vertically downwards and Os is the displacement which takes place vertically upwards. Hence the virtual work is F • Ss= — Wo(2a cos° — cot0). To explain eqn. , is of fixed length, and have inserted the forces T acting at B and D in place of the constraint. The forces T each act in the line BD, so that contributions to the scalar product F • Os will only arise from components of Ss which take place in the line BD.
Of course, if X = 0 , Y = 0 , and G(0) = 0, then the set of forces is in equilibrium and the system cannot reduce to a force or couple. Thus we have shown that, in general, an arbitrary set of forces is equivalent either to a single force R, or to a couple, or is in equilibrium. ) Later we show that two sets of forces which are equivalent, or reduce to the same R and G(0), give the same accelerations to a lamina. If two sets of forces are equivalent and all the forces in one set are reversed, then the first set is in equilibrium with the reversed set.
A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics by Brian H. Chirgwin