EXERCISES
97
--
i.
V.
T.
ii.
T
.
8.
iii.
(V
.
T)
x
V.
iv.
(V
.
T)
x
(1.
v).
v.
(V
.
T)
.
(T
.
V).
(V
.
T)
=
0
for any
V?
T)
=
0
for any
v?
(b)
What values for the elements of
r,
other than
Ti,
=
0,
will result in
V
X
(c)
What values for the elements of
T,
other
than
Tij
=
0,
will result in
V. (V.
--
-
-
16.
Use subscriptlsummation notation
to
show that the expressions
(T.C)XD
and
T.
CxD
=(
1
are not necessarily equal
17.
An orthogonal
(u,
v,z)
coordinate system
is
defined by the set of equations
relating its coordinates to a standard set of Cartesian coordinates,
X
(a)
Determine the elements of the transformation matrix
[a].
(b)
-
At the Cartesian point
(1,1,1)
let the vector
v
=
1
8,
+
2
(c)
At the Cartesian point
(1,1,1)
let the tensor
+
3
Sz.
Express
V
at this point using
(u,
u,
z)
system vector components and basis vectors.
=
1
6183
+
2
&2C1
+
3
6263
+
46362.
Express
T
at this point using
(u,
v,z)
system tensor elements and
basis vectors.
18.
Consider
a
dumbbell positioned in the xy-plane of a Cartesian system, as shown
in the figure below. The moment
of
inertia tensor for this object expressed in this
Cartesian system is
-
- -
1
=
I.
.&&
'1
1
J
-2
0
144,
[;2
;
(I].
Find the basis vectors
of
a coordinate system in which the moment
of
inertia
tensor is diagonalized. Draw the dumbbell in that system.
