Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/183

In laying out the curves on a railroad it will not do, on account of the high speed of trains, to pass abruptly from a straight stretch of track to a circular curve. In order to make the change of direction gradual, engineers make use of transition curves to connect the straight part of a track with a circular curve. Arcs of cubical parabolas are generally employed as transition curves.

Illustrative Example 3. The transition curve on a railway track has the shape of an arc of the cubical parabola $y={\tfrac {1}{3}}x^{3}$. At what rate is a car on this track changing its direction (1 mi. = unit of length) when it is passing through (a) the point (3, 9)? (b) the point $(2,{\tfrac {8}{3}})$? (c) the point $(1,{\tfrac {1}{3}})$?

Solution.	${\frac {dy}{dx}}=x^{2},{\frac {d^{2}y}{dx^{2}}}=2x$.
Substituting in (40),	$K={\frac {2x}{(1+x^{4})^{\frac {3}{2}}}}$.
(a) At (3, 9),	$k={\frac {6}{(82)^{\frac {3}{2}}}}$ radians per mile = 28′ per mile.
(b) At $(2,{\frac {8}{3}})$,	$K={\frac {4}{(17)^{\frac {3}{2}}}}$ radians per mile = 3° 16′ per mile. Ans.
(c) At $(1,{\frac {1}{3}})$,	$K={\frac {2}{(2)^{\frac {3}{2}}}}={\frac {1}{\sqrt {2}}}$ radians per mile = 40° 30′ per mile. Ans.

103. Radius of curvature. By analogy with the circle (see (38), § 99), the radius of curvature of a curve at a point is defined as the reciprocal of the curvature of the curve at that point. Denoting the radius of curvature by R, we have

	$R={\frac {1}{K}}$;[1]
Or, substituting the values of X from (40) and (41),
(42)	$R={\frac {\left[1+\left({\frac {dy}{dx}}\right)^{2}\right]^{\frac {3}{2}}}{\frac {d^{2}y}{dx^{2}}}}$;
(43)	$R={\frac {\left[\rho ^{2}+\left({\frac {d\rho }{d\theta }}\right)^{2}\right]^{\frac {3}{2}}}{\rho ^{2}-\rho {\frac {d^{2}\rho }{d\theta ^{2}}}+2\left({\frac {d\rho }{d\theta }}\right)^{2}}}$[2]

1. Hence the radius of curvature will have the same sign as the curvature, that is, + or -, according as the curve is concave upwards or concave downwards.
2. In § 98, p. 152, (43) is derived from (42) by transforming from rectangular to polar coordinates.