. Field-book for railroad engineers. Containing formulas for laying out curves, determining frog angles, levelling, calculating earth-work, etc., etc., together with tables of radii, ordinates deflections, long chords, magnetic variation, logarithms, logarithmic and natural sines, tangents, etc., etc . Fig. 21. Article IV, — Miscellaneous Problems. 65, Problem. Given A B = a [Jig. 22) and the perpendicularB C = b, to Jind the radius of a curve that shall pass through C and thetangent point A. Solution. Let 0 be the centre of the curve, and draw the radii A 0and C 0 and the line CD parallel to
. Field-book for railroad engineers. Containing formulas for laying out curves, determining frog angles, levelling, calculating earth-work, etc., etc., together with tables of radii, ordinates deflections, long chords, magnetic variation, logarithms, logarithmic and natural sines, tangents, etc., etc . Fig. 21. Article IV, — Miscellaneous Problems. 65, Problem. Given A B = a [Jig. 22) and the perpendicularB C = b, to Jind the radius of a curve that shall pass through C and thetangent point A. Solution. Let 0 be the centre of the curve, and draw the radii A 0and C 0 and the line CD parallel to A B. Then in the right triangleCOD we have 0 C^ = CD + OD^ But 0 C = R, CD = a, andOD = AO — AD = R — b. Therefore, R = a-{- {R — 6)» =a^ + R^ — 2 Rb -\- b\ or 2 Rb = a^ -{- b^; 2 b Example. Given a = 204 and b = 24, to find R. Here R »- 204-2 242X-24 + 2 = «67 + 12 = 879. iillSCELLANEOUS PROBLEMS. 47 C6. Corollary 1. If R and b are given to find A B = a, thatvs, to determine the tangent point from which a curve of given radius. most start to pass through a given point, we have (§65) 2Rb =fl«-f i^ora = 2Rb — b^; ..a = ^b {2R — b). Example. Given 6 = 24 and 72 = 879, to find a. Here o =-/94 (1758 — 24) = ^ 41616 = 204. 67. Corollary 2. If R and a are given, and b is required, wehave (§65) 2 Rb = a^ + 6^ or 6« — 2Rb = —a}. Solving thisequation, we find for the value of b here required, b = R — ^R- — a\ 68. Problem. Given the distance AC = c [Jig. 22) and the an-gle B A C ^ A, to find the radius R or deflection angle D of a curve, thatfhall pass through C and the tangent point A. Solution. Draw 0 E perpendicular to A C Then the angle AOE^^A0C = BAC=A{(j2, III.), and the right triangle A OEgWos ()^0 = 3j^^i^; • R- ^^Sin. A To find Z), we have (§ 9) sin. D =•nst found, we have sin. Z) = 50 -^• ^. Substituting for R its value hesin. A 48 CIRCULAR CURVES. c Example. Given c = and ^l = 5°, to find R and D. Heix. ^, ,^„,„ , .
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