6-39. In a rectangular coordinate system a particle moves around a unit circle in
the plane z= 0 with a constant angular velocity of ω= 5 rad /sec
(a) What is the angular velocity vector for this system?
(b) What is the velocity of the particle at any time tif the position of the particle
is
r = cos 5tˆe 1 + sin5 tˆe 2?6-40. A particle moves along a curve having the parametric equations
x=et, y = cos t, z = sin t.(a) Find the velocity and acceleration vectors at any time t.
(b) Find the magnitude of the velocity and acceleration when t= 0.
6-41. Let x=x(t), y =y(t)denote the parametric representation of a curve in
two-dimensions. Using chain rule differentiation, show that the center of curvature
vector, at any parameter value t, can be represented by
c(t) = x(t)ˆe 1 +y(t)ˆe 2 +x ̇(^2) + ̇y 2
x ̇y ̈−y ̇x ̈
(−y ̇ˆe 1 + ̇xeˆ 2 )
provided x ̇y ̈−y ̇ ̈xis different from zero. Here the notation x ̇=dxdt and x ̈=d
(^2) x
dt^2 has
been employed.
6-42. Find the center and radius of curvature as a function of x for the given
curves.
(a) (x−2)^2 + (y−3)^2 = 16 (b) y=ex6-43. Let e denote a unit vector and let A denote a nonzero vector. In what
direction e will the projection A·e be a maximum?
6-44. Assume A=A(t)and B =B(t).
(a) Show that d
dt(
A·B)
=A·d
B
dt+d
A
dt·B(b) Show that
d
dt(
A×B)
=A×dB
dt +dA
dt ×
B6-45. Given A=t^2 ˆe 1 +teˆ 2 +t^3 ˆe 3 and B = sin tˆe 1 + cos tˆe 2 +ˆe 3.
Find (a)
d
dt(
A·B)
(b)d
dt(
A×B)
(c)d
dt(
B·B)