A Course of Mathematics for Engineers and Scientists. Volume by Brian H. Chirgwin, Charles Plumpton

By Brian H. Chirgwin, Charles Plumpton

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Extra resources for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics

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Iii) A uniform rectangular lamina ABC D of edges AD = 2a, AB = 2b, has two smooth hinges at points P (0, c, 0) Q (0, — c, 0) on the horizontal edge AD, the mid-point of AD being the origin 0 of coordinates, 0z being vertically upwards and Oy along the hinged edge. The lamina is kept at an angle a with the vertical plane x = 0 by a light string joining a fixed point H, (0, 0, h), to a point K of the lamina distant p from AD and d from OG on the side nearer P, where G is the centre of mass, and a smooth stop at D prevents the lamina sliding off the hinges.

Since CN = 3GN, CG = 2GN, this equation gives 6R3= 4 (W + R3 = W + + (2 W + + + W2). + W3. § 2 : 2] 53 SETS OF FORCES : EQUILIBRIUM Similarly, R, = W + g w, + w2 + 1472, R2 = w + + w2 + W3 Effectively, we have used three conditions of equilibrium by taking moments about three different axes. No fresh information can be obtained by resolution since R1 ± R2 + R3 = 3 W + + W2+ W3. Next we consider the equilibrium of one of the rods, OA, again by taking moments about suitable axes through 0. Since all the forces on OA except T2 and T3 have no moment about a vertical axis through 0, we deduce that T2sink/6 = T3sink/6.

We take moments about H so that the reso lutes of T do not occur in the equations. Then, since H is the point (0 0 h}, F (H) = {b sina 0 b cosoc — h) x {0 0 — W} + {0 c — 74 x {X1 0 Z1} + + (0 — c —h) x {X2 0 Z2) + (0 — a —h) x (0 Y 0) = 0. (1) The other three conditions of equilibrium are, from F = 0, Xi + X2 + Ti = 0, Y + T2 = 0 , - W + Zi + Z2 + 11, = O. (2) (3) (4) These equations are obtained simply by adding the corresponding resolutes in the columns of specification of the forces above. The conditions of equilibrium obtained from the resolutes of (1) are : cZ„ — cZ2 + h Y = 0, Wbsincc —hX,— hX2 = 0, — cX„ + cX2 = O.

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