Calculus: Analytic Geometry and Calculus, with Vectors

7.3 Center and radius of curvature 437

where n is the unit normal obtained by rotating t clockwise through the angle
a/2, ds/dt is the speed of the particle, and K is the curvature dO/ds of C. Show
that writing the formula v = IvIt in the form

(3)

and differentiating give

V_ds

Wt t

2

(4) a = at =ate t + as at

and that use of (2) then gives

(5) a - atit+(dt)2Kn.

Remark: This elegant formula gives the normal (or transverse) and tangential

scalar components of the acceleration in terms of the speed and rate of change of

the speed of the particle. The simplest applications involve the case in which the

particle moves along the curve with constant speed v so that ds/dt = a. and

d2s/dt2 = 0 at each time. In this case, (5) reduces to the simple but important

formula

(6) a = v2Kn.

If K = 0, then a = 0. If K 0 0, then a has magnitude a21 K1 or a2/p (where p

is the radius of curvature) and is directed toward the center of curvature. Use of

(6) and the formula F = ma gives the force required to propel a particle of mass m

along a curve C with constant speed v.

18 Let C be a simple closed convex curve which could, for example, be a

famous "triangular roller" composed of the three vertices of an equilateral tri-

angle together with three circular arcs of which each has its center at one vertex

and contains the other two vertices. Let R be a rod (or line segment) whose

length L is small enough to permit the rod to be "slid around" C in such a way

that its two ends both remain on C. It is easy to presume, as has sometimes

been done in "proofs" of the "Holditch theorem," that each point P on R must

traverse a simple closed convex curve Cr as R slides around C. Persons having

a compass, a straightedge like the edge of a sheet of paper upon which marks

can be placed, and some spare time at their disposal can sketch interesting

figures. Surprises await those who let C be a triangular roller, let the length L

of the rod R be equal to or only a bit less than the distance between two vertices

of the equilateral triangle, and let P be the mid-point of L.