Calculus: Analytic Geometry and Calculus, with Vectors

182 Functions, limits, derivatives

current I in the circuit containing the capacitor. Find a formula which gives I

in terms of t.

9 This problem involves uniform circular motion. Let a particle P start

at the point (a,0) of the plane Figure 3.791 and move around the circle with

center at the origin and radius a in such a way that

the vector OP rotates at the constant positive rate

co (omega) radians per second. Letting r = pp,

show that

(1) r = a(cos wti + sin wtj)

(2) v = aco(-sin wti + cos wtj)

(3) a = -aw2(cos wti + sin wtj)

and hence that

Figure 3.791 (4) a = - aw2u,

where, at each time t, u is a unit vector running from the origin toward P.

Show that 0 and interpret this result. Show that Ivi = aw and interpret

this result. Remark: The result (4) is important in physics. It says that, in

uniform circular motion, the particle is always accelerated toward the center and

that the magnitude of the acceleration is awe. Some additional terminology

should be encountered frequently and slowly absorbed. When a particle moves

upon a line in such a way that its coordinate at time t is el + B sin (wt + ¢),

the motion is said to be sinusoidal or (particularly in old books) harmonic or

simple harmonic. The numbers 95, w/2r, and B are the phase, the frequency

(cycles per unit time), and the amplitude of the motion. Glances at the compo-

nents of r and a in the above formulas show that the projection of P upon a

diameter (line, not number) of the circle executes sinusoidal motion. More-

over, the projection is always accelerated toward the center, and the magnitude

of the acceleration is proportional to the distance from the center. See also

Problem 16.

10 As in the text of this section, let a particle move in E3 in such a way that

its displacement, velocity, and acceleration are

(1) r(t) = x(t)i + y(t)j + z(t)k

(2) v(t) = x'(t)i + y'(t)j + z'(t)k

(3) a(t) = x"(t)i + y"(t)j + z"(t)k

and the square of its speed is

(4) IV(t)12= [x'(t)]2 + [y'(t)]2 + [z'(t)]2.

Using this information, prove that if the particle moves with constant speed c,

then the acceleration is always orthogonal to the velocity. Hints: Do not get

scared. Equate the right member of (4) to c2. Equate the derivatives of the

members of your equation. Look at your result. Remark: One who thinks that

this result is mysterious should remember or discover which way he tends to

topple when he sits in an automobile which rounds an unbanked curve at con-

stant speed.