By Brian H. Chirgwin
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Extra info for A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics
If AB, BC, CD make acute angles 0, q), y) with the vertical, and CE is not greater than BF, prove that the thrusts in BF, CE are respectively W (5 tan 3 tan cp) , z W (tan 1p — 3 tan 9)) . Find the reaction at the joint D. 11. Five equal uniform rods AB, BC, CD, DE, EA, each of weight W and length 2a, are smoothly jointed together at A, B, C, D, E and rest in equilibrium in a vertical plane with the joint A uppermost, and AB, AE equally inclined to the vertical and in contact with two smooth fixed pegs, on the same horizontal level and distant 2 c (< 2a) apart.
But as in example (ii) above BD = yB = c secyB . Hence the total length of the string ABD is c sec yB + 2c tanyB. But this is given as 11a and so yBsatisfies the equation (1) secyB + 2 tanyB = 11/4. h. side of (1) is a strictly increasing function of yB as yBincreases from 0 to an and hence (1) has one root only in this range, given by tanyB = 1 :1as required. (secy B tantpB) = 8a log2. (v) A uniform chain of length 2/, and of weight 2w/ hangs in a vertical plane with one end 0 fixed and the other end A attached to a uniform rod AB of weight w 1.
Prove that the system is equivalent to a single force in the line Xy — Yx G 0. The moments of a system of coplanar forces about the points (2, 1), ( 3, 4) and (1, — 3) are 11, — 15 and 15 units respectively. Find the magnitude and the equation of the line of action of the force acting at the point (3, 2) which, together with the given system, constitute a couple. Show that the moment of the couple is 13 units. — The moment of the given system (X, Y) and 0 about the point (x, y) is k0' = kG — (ix + jy) X (iX --F j Y) = (0 — x Y + yX)k.
A Course of Mathematics for Engineers and Scientists. Volume 3: Theoretical Mechanics by Brian H. Chirgwin