Calculus: Analytic Geometry and Calculus, with Vectors

7.2 Lengths and integrals 423

5 When 0 < b < a and a circle (or cog wheel) of radius b rolls without
slipping inside a circle of radius a as in Figure 7.291, the path traced by a point
on the small rolling circle is a hypocycloid (inside-cycloid) that appeared in one
of Problems 6.59 and that even small dic-
tionaries describe. It can be shown that
when the center of the small circle has
passed over an angle 0 (that could be wt),
the point initially at (a,0) has coordinates

a-b

x=(a-b)cos0+bcos b 0

y=(a-b)sin0-bsinab b0.

With the aid of Figure 7.291 find the

Figure 7.291

amount by which 0 must increase as P traverses one arch of the cycloid from the

big circle back to the big circle. Then find the length of one arch of the cycloid.

tins.: 8b(a - b)/a.

6 It is a remarkable fact that when b = a/4, so that the hypocycloid of the

preceding problem has four cusps, the equations can be put in the form

x =

4

(3 cos 0 + cos 30) = a cos' 0

y=4(3sin0-sin30)=asin30

so that

x34 + y35 = a%.

Using this formula, find the length of the part of the hypocycloid in the first

quadrant. .4ns.: 3a12.

7 Let f and g have continuous derivatives over a <_ u < b. Let C be the

curve which the pointP(f(u), g(u)) traverses in an xy plane as u increases from

a to b. Let C be regarded as a wire having linear density S so that a piece of C

of length L has mass SL. Let p be a nonnegative integer and let xo be a constant.

Set up an integral for the pth moment of the wire about the line having the

equation x = x0.

8 With or without assistance from the preceding problem, set up an integral

for the pth moment about a diameter of a circular wire having linear density S.

9 Centroids of triangles, rectangles, and regular polygons coincide with the

centroids of the regions that they bound.

Draw some polygons for which there is violent

departure from coincidence.

(^10) Figure 7.292 is intended to steer our
thoughts toward a wire or cord concentrated
on a curve C and to make us realize that we
have not yet calculated the gravitational
force F upon a particle P* of mass m at
P(x,y,z) that is produced by the wire. Set
up an integral for F. Outline of solution:
Since x, y, z have been preempted, we take
Figure 7.292
Z
Y