Page 26 - Dynamics of Mechanical Systems

0593_C01_fm Page 7 Monday, May 6, 2002 1:43 PM

Introduction 7

Consider the task of locating P relative to the origin O by moving from O to P along
lines parallel to X, Y, and Z, as shown in Figure 1.6.3. The coordinates may then be
interpreted as the distances along these lines. The distance d from O to P is then given by
the Pythagorean relation:

d = ( x + y + ) 12 (1.6.1)
/
2
2
2
z
Finally, the point P is identified by either the name P or the set of three numbers (x, y, z).
To illustrate the use of additional lines and angles, consider the cylindrical coordinate
system shown in Figure 1.6.4. In this system, a typical point P is located relative to the
origin O by the coordinates (r, θ, z) measuring: (1) the distance r along the newly introduced
radial line, (2) the inclination angle θ between the radial line and the X-axis, and (3) the
distance z along the line parallel to the Z-axis, as shown in Figure 1.6.4.
By comparing Figures 1.6.3 and 1.6.4 we can readily obtain expressions relating Cartesian
and cylindrical coordinates. Specifically, we obtain the relations:

x = cosθ
r
y = sinθ (1.6.2)
r
z = z

and

r = ( x + ) 12
/
2
2
y
yx)
θ = tan −1 ( / (1.6.3)
z = z
As a third illustration, consider the coordinate system shown in Figure 1.6.5. In this
case, a typical point P is located relative to the origin O by the coordinates (ρ, φ, θ)
measuring the distance and angles as shown in Figure 1.6.5. Such a system is called a
spherical coordinate system.

Z Z
P(x,y,z) P(r, ,z)

O z O z
Y Y
x θ

y
X X
FIGURE 1.6.3 FIGURE 1.6.4
Location of P relative to O. Cylindrical coordinate system.