Page 50 - Standard Handbook Of Petroleum & Natural Gas Engineering

Differential and Integral Calculus 39

expresses the rate of change of u with respect to t, in terms of the separate
rates of change of x, y, . . . with respect to t.
Radius of Curvature

The radius of curvature R of a plane curve at any point P is the distance along
the normal (the perpendicular to the tangent to the curve at point P) on the
concave side of the curve to the center of curvature (Figure 1-33). If the equation
of the curve is y = f(x)

where the rate of change (ds/dx) and the differential of the arc (ds), s being
the length of the arc, are defined as

and

ds = Jdx' + dy2

and dx = ds cos u
dy = ds sin u
u = tan-'[f'(x)]

u being the angle of the tangent at P with respect to the x-axis. (Essentially,
ds, dx, and y correspond to the sides of a right triangle.) The curvature K is
the rate at which <u is changing with respect to s, and

1 du
K=-=-
R ds
If f'(x) is small, K s f"(x).

Figure 1-33. Radius of curvature in rectangular coordinates.