Page 254 - Dynamics of Mechanical Systems

0593_C07_fm Page 235 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 235

P7.4.7: See Problems P7.2.5 and P7.4.4. Find the second moments of S relative to O for the
directions of n , n , and n .
z
x
y
P7.4.8: See Problems P7.3.3 and P7.4.4. Find the following moments and products of inertia
SO
of S for O: I SO , I SO , I SO , I SO , and I .
xx xz yy yz zz
P7.4.9: See Problems P7.2.6 and P7.4.4. Let n and n be unit vectors with coordinates
b
a
relative to n , n , and n as:
z
x
y
n = 0 75 n − 0 5 n + 0 433 n
.
.
.
a x y z
n = 0 655 n + 0 756 n
.
.
b y z
SO
Find the second moment vectors I SO and I .
a b
P7.4.10: See Problems P7.2.5, P7.2.6, P7.3.7, P7.4.4, and P7.4.9. Let n and n be the unit
a
b
vectors of Problem P7.4.9. Find the following moments and products of inertia of S relative
SO
to O: I SO I , SO , and I .
aa ab bb
Section 7.5 Transformation Rules
P7.5.1: Let S be a set of eight particles P (i = 1,…, 8) located at the vertices of a cube as
i
in Figure P7.5.1. Let the masses m of the P be as listed in the figure. Determine the second-
i
i
moment vectors I SO I , SO , and I S O for the directions of the unit vectors n , n , and n shown
1 2 3 1 2 3
in Figure P7.5.1.
Z n
3
P
P 4 3 m = 2 kg, m = 4 kg
1 5
m = 3 kg, m = 6 kg
2 6
2m m = 1 kg, m = 3 kg
P 3 7
P 7
8 m = 5 kg, m = 2 kg
4 8
P P 2
1
Y
O
n
2m 2
P 5
P
2m 6
X n
1
FIGURE P7.5.1
Particles at the vertices of a cube.
P7.5.2: See Problem P7.5.1. Let n , n , and n be unit vectors with components relative to
a
b
c
n , n , and n as:
3
2
1
.
.
n = 05 n + 0866 n
a 1 2
.
n =−0 433 n + 0 25 n + 0 866 n
.
.
b 1 2 3
.
.
n = 075 n − 0 433 n + 05 n
.
c 1 2 3