Calculus: Analytic Geometry and Calculus, with Vectors

184 Functions, limits, derivatives

13 While the matter must remain mysterious until some mathematical
secrets have been revealed, the tip P of a cog of a particular hypocyclic gear
moves in such a way that its displacement vector at time t is
r = a(cos3 wti + sin3 wtj),
where a and w are positive constants. Find and simplify formulas for its velocity,
speed, and acceleration. Ans.:

v =

Saw

sin 2wt(-cos wti + sin wtj)

Speed =32w sin 2wtI

3aw2

a = 2 sin 2wt(sin cod + cos wtj) + 3aw2 cos 2wt(-cos cod + sin wtj).

14 A particle P moves in such a way that its displacement vector at time t is

_ 2t t2-1,

Show that Iri = 1 at all times and hence that the path of P must lie on the unit

circle with center at the origin. If a particle moves on a circle in such a way that

it has a nonzero velocity vector v, then v must be tangent to the circle and hence

orthogonal (or perpendicular) to r. Check this story by calculating v and show-

ing that yr = 0. Find the times at which the particle crosses the coordinate

axes, and then obtain more information about the motion of P.

15 Prove that if a particle P moves in E3 in such a way that it has displace-

ment vectors and velocity vectors r(t) and v(t) at time t, then

d

jr(t)l=

r

dt jr(t)lt)

when the particle is not at the origin. Tell why this implies that if the path of P

lies on a sphere with center at the origin, then v must be normal (or orthogonal

or perpendicular) to r and hence v must be tangent to the sphere. Remark:

It can be presumed that we do not known much about curves and surfaces in

Es, but we can presume that if a particle P makes a decent trip along a decent

curve lying on a decent surface, then at each time the velocity vector having

its tail at P must be tangent to the surface as well as to the curve.

16 If a particle moves along the x axis in such a way that, at time t,

x = a sin (cot + 0)

where a, co, 0 are constants for which a > 0 and co > 0, the particle is said to

describe (or execute) sinusoidal (or harmonic) motion. Calculate the first and

second derivatives of x with respect to t and show that

d2x

dt2 = - w2x.

Remark: This shows that the scalar acceleration of the particle is proportional to

the scalar displacement of the particle from the origin (or equilibrium position)

about which it oscillates.