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0593_C07_fm Page 230 Monday, May 6, 2002 2:42 PM

230 Dynamics of Mechanical Systems

where we have made use of Eqs. (7.12.4) and (7.12.5). Similarly, the second term of Eq.
(7.12.2) may be expressed as:

 N 
ωω ×  ∑ m r ×( ωω × ) = ωω ×(I BG ⋅ ωω) (7.12.7)
r
ii
i
  = i 1  
*
Hence, T becomes:
ωω
T =− I BG ⋅ −αωω × I ( BG ⋅ ) (7.12.8)
α
*
The scalar components of T are sometimes referred to as Euler torques. We will explore
*
the significance of these terms in the next several chapters.

References
7.1. Kane, T. R., Analytical Elements of Mechanics, Vol. 2, Academic Press, New York, 1962.
7.2. Kane, T. R., Dynamics, Holt, Rinehart & Winston, New York, 1968.
7.3. Kane, T. R., and Levinson, D. A., Dynamics: Theory and Applications, McGraw-Hill, New York,
1985.
7.4. Huston, R. L., Multibody Dynamics, Butterworth-Heinemann, Stoneham, MA, 1990.
7.5. Hinchley, F. A., Vectors and Tensors for Engineers and Scientists, Wiley, New York, 1976.
7.6. Hsu, H. P., Vector Analysis, Simon & Schuster Technical Outlines, New York, 1969.
7.7. Haskell, R. E., Introduction to Vectors and Cartesian Tensors, Prentice Hall, Englewood Cliffs, NJ,
1972.

Problems

Section 7.2 Second Moment Vectors

P7.2.1: A particle P with mass 3 slug has coordinates (2, –1, 3), measured in feet, in a
Cartesian coordinate system as represented in Figure P7.2.1. Determine the second
moment of P relative to the origin O for the directions represented by the unit vectors n ,
x
n , n , n , and n , where n and n are parallel to the X–Y plane, as shown in Figure P7.2.1.
b
a
a
b
y
z
That is, find I PO I , PO I , PO I , PO , and I PO .
x y z a b
Z
n z Q(-1,2,4)
P(2,-1,3)
n b
O 30° Y
n y
45°
FIGURE P7.2.1
n x n
A particle P in a Cartesian reference a
frame. X