By Brian H. Chirgwin
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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics
12. information provided are together sufficient to give unique answers to the problems concerned. We now give examples in which it is impossible to find all the forces uniquely and which correspond to situations of very common occurrence in practice. Consider a rigid rod AB pinned at each end and carrying a weight attached at a point of its length, Fig. 12. ) It is impossible, by the methods of statics alone, to determine uniquely the forces acting on this system. The conditions of equilibrium furnish three linear equations between the four unknown forces P, Q, R, S.
If the angle between the forces is given, show that, whatever their directions, the resultant B of P and Q passes through a fixed point. If A and B have coordinates (a, 0) and (0, b) referred to fixed rectangular axes in the plane, and if P is in the positive direction of the xlaxis when Q is in the positive direction of the y-axis, show that the coordinates of the fixed point are x = P(aP bQ)1B2, y = Q(aP bQ)IR2 • 4. The points P, Q, R on the sides BC, CA, AB respectively of a triangle ABC divide each of these sides in the ratio 1+ k:1—k.
Prove that when a weight 4 W is suspended from B the tension in the string is 2 1/3 W. 4. Six rods are freely jointed at their ends to form a hexagon ABCDEF with AB = CD = DE = FA = 2a each of weight W, , and BC = EF = 2 b each of weight W2 . The joints B and F and the joints C and E are joined by light struts and the framework is freely suspended from A. It is in equilibrium in a vertical plane with the struts BF, CE horizontal. If AB, BC, CD make acute angles 0, q), y) respectively with the vertical, and CE is not greater than BF, find the thrusts in CE and BF.
A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics by Brian H. Chirgwin