Page 27 - Dynamics of Mechanical Systems

0593_C01_fm Page 8 Monday, May 6, 2002 1:43 PM

8 Dynamics of Mechanical Systems

By comparing Figures 1.6.3 and 1.6.5, we obtain the following relations between the
Cartesian and spherical coordinates:

φ
x = ρsin cos θ
φ
y = ρsin sin θ (1.6.4)
z = ρcos φ

and

/
2
ρ = (x 2 + y 2 + ) 12
z
/
2
φ = cos −1  z/ (x 2 + y 2 + ) 12  (1.6.5)
z
   
θ = tan −1 (yx/ )
The uses of vectors and coordinate systems are closely related. To illustrate this, consider
again the Cartesian coordinate system shown in Figure 1.6.6. Let the unit vectors n , n ,
y
x
and n be parallel to the X-, Y-, and Z-axes, as shown. Let p be a position vector locating
z
P relative to O (that is, p is OP). Then it is readily seen that p may be expressed as the
vector sum (see details in the next chapter):

p = x n + y n + z n (1.6.6)
x y z
Also, the magnitude of p is:

/
2
p = (x 2 + y 2 + ) 12 (1.6.7)
z

Z Z
Π(ρ,∅,θ)
n
z P(x,y,z)
∅
p

O
Y O
Y
θ n
n y
x
X X

FIGURE 1.6.5 FIGURE 1.6.6
Spherical coordinate system. Position vector in a Cartesian coordinate system.