By Brian H. Chirgwin
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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics
11. A polygon A,... An(not necessarily regular) is made up of 92 rods A,A„ An ,A„, Anil„ 44 A COURSE OF MATHEMATICS freely jointed at their ends, and placed on a smooth horizontal table. Each rod experiences a force proportional to its length, applied at its mid-point, and directed along the outward normal. Show that in equilibrium the joints A1, An lie on a circle, and that the reaction at each joint is the same. If all the rods are of the same length, and the common force applied to each of them is F, calculate the reaction at the joints.
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.
IL V (P) = —I i Qi dqi . 19) A This is the general definition of potential energy and can be expressed in words thus : "A set of forces acting on a system is conservative if the net work done by an agency in displacing the system from a configuration A to an arbitrary configuration P is independent of the 'path' followed by the system from A to P. " This definition implies that the potential energy is arbitrary to the extent of an additive constant ; an alteration of the 'standard' position A adds a constant, independent of P, to the expression for V (P) given in eqn.