Page 253 - Dynamics of Mechanical Systems

0593_C07_fm Page 234 Monday, May 6, 2002 2:42 PM

234 Dynamics of Mechanical Systems

where n , n , and n are mutually perpendicular unit vectors. Compute the following
1
2
3
dyadic products: (a) ab, (b) ba, (c) ca + cb, (d) c(a + b), (e) (a + b)c, and (f) ac + bc.
P7.4.2: See Problem 7.2.1. A particle P with mass 3 slug has coordinates (2, –1, 3), measured
in feet, in a Cartesian coordinate system as represented in Figure P7.4.2. Determine the
inertia dyadic of P relative to the origin O, I P/O . Express the results in terms of the unit
vectors n , n , and n .
x
z
y
Z
n z Q(-1,2,4)

P(2,-1,3)

O Y
n y
FIGURE P7.4.2
n x
A particle P in a Cartesian reference
frame. X

P7.4.3: See Problem P7.2.2. Let Q have coordinates (–1, 2, 4). Repeat Problem P7.4.2 with
Q instead of O being the reference point. That is, find I P/Q .
P7.4.4: See Problems P7.2.5 and P7.3.3. Let S be the set of three particles P , P , and P
1 2 3
located at the vertices of a triangle as shown in Figure P7.4.4. Let the particles have masses:
2, 3, and 4 kg, respectively. Find the inertia dyadic of S relative to O, I S/O . Express the
results in terms of the unit vectors n , n , and n .
x y z

Z
P (2,2,4)
n z 2

P (0,5,2)
3
O P (1,1,1) Y
1
n
y
FIGURE P7.4.4 (units in meters)
Particles at the vertices of a triangle. X n x

P7.4.5: See Problems P7.2.8, P7.2.9, P7.3.4, and P7.4.4. Let G be the mass center of S. Find
the inertia dyadic of S relative to G, I . Express the results in terms of the unit vectors
S/G
n , n , and n .
x
z
y
P7.4.6: See Problems P7.4.4 and P7.4.5. Let G have an associated mass of 9 kg. Find the
inertia dyadic of G relative to the origin O, I G/O . Express the result in terms of the unit
vectors n , n , and n .
y
x
z
P7.4.7: See Problems P7.4.5 and P7.4.6. Show that:
I SO = I S G + I G O