This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1877 edition. Excerpt: ...line. Therefore, if the centre of gravity g be moved along the line OB, and the end E be kept constantly against the vertical BE, the end A will trace the curve. But this is equivalent to the well-known method of constructing an ellipse by means of a trammel. Hence, the curve CAD is an ellipse. Let AE= ...
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1877 edition. Excerpt: ...line. Therefore, if the centre of gravity g be moved along the line OB, and the end E be kept constantly against the vertical BE, the end A will trace the curve. But this is equivalent to the well-known method of constructing an ellipse by means of a trammel. Hence, the curve CAD is an ellipse. Let AE= 1 and Ag = l; x = BF, y = FA; then, Eg = tx, and FAi+Fg=Ag or, tf+lx = IP;.which is the equation of the ellipse. A more general solution will be found by letting Ag = n. Eg. 238. Required the form, of a curve such that a heavy bar restino against it and against a smooth pin above the curve, will be in equilibrium in all positions. Let AD be the bar, D the position of the pin, and ABC the required curve. This is also a case in which the potential energy is constant; hence, the centre of gravity will be found in a horizontal line gg', passing through the centre of gravity, g, of the bar in the vertical position. The curve may therefore be constructed by drawing any number of radial lines DB, DA, etc., through D, intersecting them by the horizontal line gg', and laying off on the radial line.s below the horizontal the constant distance gB = g-A, etc. The curve is called the conchoid of Nicomedes. Let Ag-= Bg = a; gD = e; ADB = 8; AD = p; then, AD = Ag-+g-D = Ag-+ Dg sec 0; or, P = a + c sec 8 which is the polar equation of the curve. 239. A cord of given length is suspended at two points in the same horizontal; required the form of the curve when the centre of gravity is the lowest. The cord, being perfectly flexible, will naturally assume the position of stable equilibrium, and its potential energy will be a minimum; that is, its centre of gravity will be the lowest possible. The curve assumed by such a cord is called a Catenary. If the cord...
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