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v = ɺɺ
a = ɺ x x
x
a = ɺ y y
v = ɺɺ 2-15
y
a = ɺ z z
v = ɺɺ
z
,
,
here aa a represent, respectively, the first time derivatives of
x
z
y
(), v =
() or the second time derivatives of the
(), v =
v = vt y vt z vt
x
y
z
x
(), z =
functions x = xt yt zt
().
(), y =
The acceleration has a magnitude
2
2
2
a= a + a + a 2-16
x
y
z
( )
( )
and a direction specified by cosines cos , = a x , cos , =aj a y ,
ai
a a
a
cos ( ,ak )= z , or the unit vector u = a / a .
a a
41 Motion of Particle: Cylindrical Components
Sometimes the motion of the particle is constrained on a path
that is best described using cylindrical coordinates. If motion is
restricted to the plane, then polar coordinates are used.
Polar Coordinates. We can specify the location of the particle
shown in Fig. 2-6 using a radial coordinate r, which extends outward
from the fixed origin O to the particle, and a transverse coordinate θ
which is the counterclockwise angle between a fixed reference line
and the r axis. The angle is generally measured in degrees or radians,
where 1rad = 180 / π° . The positive directions of the r and θ
coordinates are defined by the unit vectors u and u respectively.
r
θ
Here u is in the direction of increasing r when θ is held fixed, and u θ
r
is in a direction of increasing θ when r is held fixed. Note that these
directions are perpendicular to one another.
Position. At any instant the position of the particle, Fig. 2-6, is
defined by the position vector
r = ru (2-17)
r
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