Calculus: Analytic Geometry and Calculus, with Vectors

186 Functions, limits, derivatives

19 The vector formula

r = (b + a cos 0)cos fi + (b + a cos O)sin Oj + a sin Ok

of Problem 22 at the end of Section 2.2 provides the possibility of studying curves

on a torus. Supposing that 0 and 0 are differentiable functions of t, find r'(t).

Rns.:

r'(t) = aO'(t)[- sin 0 cos Oi - sin 0 sin q5j + cos Ok]

+ (b + a cos sin q5i + cos (hj].

20 The rod OP of the linkage of Figure 3.793 has length a and has one end

P fixed at the origin 0. The rod QP has length b.

Its lower end moves to and fro on the x axis in

such a way that its x coordinate is c + sin it at

time t. Its upper end is fastened to the first rod

O c+sin wt w Q at P, and the motion of Q causes the first rod to

Figure 3.793 rotate.

Write the formula (the law of cosines)

which expresses b2 in terms of other quantities, and

differentiate to obtain a formula for dd/dt. Then use the formula

r = OP = a (cos 0i + sin 0j)

to obtain a formula for the velocity v of P. Ins.:

v = aw

[a cos 0 - (c + sin wt)] cos wt (-sin

a(c + sin wt) sin 0 0i + cos 6j).

21 A cam can furnish us something to differentiate. A circular disk of radius

a is mounted on a cam shaft at 0.

Supposing that C is the center of the

disk and that 0 < b <= a, let IOCj = b.

The eccentric disk rotates about 0 with

constant angular speed, the angle POC

being wt. The mechanism to the right

f the dik i Fi 3 794 ke the

Figure 3.794

P(x,0) o s n gure. eps

point P(x,0) of a rod pressed against

the rotating disk so that P(x,0) moves

to and fro on the x axis as the disk rotates. The formulas

(1) OB = b cos wti, BC = b sin wtj, j PI = N/a2 -- IBCI2

show that

(2) r = [b cos wt + a2 - b2 sin2 wt]i,

where r = OP = OB + BP. Find the velocity v and the acceleration a of P,

and do not spend all day trying to discover the significance of the fact that (2)

reduces to

(3) r = a[cos wt + Icos wti]i

when b = a.