Problem #1: ( 40 points) A rectangular slab of mass
m=2500kg
is supported by cables
DA,DB
, and
DC
, which are connected to a shorter cable at point
D
attached to a crane hook as shown in Figure 1. The center of mass at point
G
of the slab is located directly below point
D
at the center of the rectangle. Use a right-handed
x-y-z
coordinate system (not drawn on the figure) that has its origin at point
G
, the positive
z
-axis directed from point
G
to
D
, and the positive
y
-axis directed from point
G
to
C
.
a) Find the unit vector
u_(as)
(directed from point
D
to point
A
), the unit vector
u_(cs)
(directed from point
D
to point
B
), and the unit vector
u_(\infty c)
(directed from point
D
to point
C
). Express the results in terms of the unit vectors
i,j
, and
k
.
b) Express the force
T_(as)
in cable
DA
, the force
T_(se)
in cable
DB
, and the force
T_(ax)
in cable
DC
in terms of the tensions
T_(in),T_(in)
, and
T_(in)
and the unit vectors
i,j
, and
k
.
c) Draw a free body diagram of point
D
, and without solving them, find the three scalar equations of equilibrium that determine the tensions
T_(at),T_(ce)
, and
T_(cc)
.