By Brian H. Chirgwin
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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics
In the critical case when the string is about to break at B, F = w11/3 Then the horizontal resolute at A gives F = To =we whence c = 1V3 since A is the vertex of the catenary. 50 A COURSE OF MATHEMATICS The equation sB = c tanyBgives tanyB= 1/1/3 so that yB= n/6. , the horizontal and vertical projections of AB, are given respectively by x B— x A = c log (secn/6 tanx/6) = /1/3 log3, yB -YA= c(secn/6 — 1) = (2 — I/3)1. (iv) ABD is a rope, of length lla and of weight w per unit length, with one end tied to a small ring A free to slide on a smooth fixed horizontal rail.
In statical problems, for example, this may take the form of saying that the rod is supported at B on a smooth support. This gives the additional relation that S = 0 42 A COURSE OF MATHEMATICS if the support has a horizontal surface, or that R = — S cota if the surface of the support makes an angle a with the horizontal. It may be a better approximation to physical arrangements to assume that the rod is not rigid. The rod therefore departs slightly from the strict linear shape, the deflection at a point depending on the stresses there and the elastic properties.
A uniform rod AB of weight W and length 2/ has one end A in contact with a smooth horizontal plane and it rests against a small smooth peg C at a distance n l from the plane (n < 2). A horizontal force P is applied at the end B in a vertical plane through AB. Find the work done by the forces acting on the rod when the angle 0 between AB and the horizontal is increased by a small amount 60. If the rod is in equilibrium when W = P1/3 and 0 is 30°, find the value of n. 2. A framework consists of four equal uniform rods, each of weight W, freely jointed at their ends to form a square ABCD, which hangs freely in equilibrium from the point A, being kept in shape by an endless light inextensible string passing through three light smooth rings, one fixed at C and the other two fixed at X and Y, the mid-points of AB and AD respectively.
A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics by Brian H. Chirgwin