PROBLEM 5.136 (Continued)
Then 50 30
00
11
1
44550
EL
x
x
dV x z dx dz
30
2
50 32
00
50
0
11
4 2 135 100
1(650 9 )
4
xz
x
xdz
zdz
PROBLEM 5.137
Locate the centroid of the plane area shown.
SOLUTION
2
,inA
,inx
,iny
3
,inxA
3
,inyA
PROBLEM 5.138
Locate the centroid of the plane area shown.
SOLUTION
2
,mmA
,mmx
,mmy
3
,mmxA
3
,mmyA
PROBLEM 5.139
A uniform circular rod of weight 8 lb and radius 10 in. is attached to a pin at C
and to the cable AB. Determine (a) the tension in the cable, (b) the reaction at C.
SOLUTION
For quarter circle,
2r
r
(a)
2
0: 0
C
r
MWTr
PROBLEM 5.140
Determine by direct integration the centroid of the area shown. Express
your answer in terms of a and h.
SOLUTION
3
(1 )yh kx
For
,0.xay
3
0(1 )hka
3
1
ka
PROBLEM 5.141
Determine by direct integration the centroid of the area shown.
SOLUTION
PROBLEM 5.142
The escutcheon (a decorative plate placed on a pipe where
the pipe exits from a wall) shown is cast from brass.
Knowing that the density of brass is 8470 kg/m
3
,
determine the mass of the escutcheon.
SOLUTION
The mass of the escutcheon is given by
(density) ,mV
where V is the volume. V can be generated by
rotating the area A about the x-axis.
From the figure:
22
75 12.5 73.9510 m
L
PROBLEM 5.142 (Continued)
Then
3
3
2
2 (3599.7 mm )
22,618 mm
VyA
PROBLEM 5.143
Determine the reactions at the beam supports for the given loading.
SOLUTION
I
I
(200 lb/ft)(15 ft)
3000 lb
R
R
PROBLEM 5.144
A beam is subjected to a linearly distributed downward load
and rests on two wide supports BC and DE, which exert
uniformly distributed upward loads as shown. Determine
the values of w
BC
and w
DE
corresponding to equilibrium
when 600
A
wN/m.
SOLUTION
PROBLEM 5.145
A tank is divided into two sections by a 1 1-m square gate that is
0.1 m
/min and the other side is filled simultaneously with methyl
alcohol (density
ma
789 kg/m
3
) at the rate of 0.2 m
3
/min,
determine at what time and in which direction the gate will rotate.
SOLUTION
Consider the free-body diagram of the gate.
First note
base
VA d
and
.Vrt
Then
3
0.1 m / min (min) 0.25 (m)
(0.4 m)(1 m)
W
t
dt
PROBLEM 5.145 (Continued)
Simplifying 32
1264.45 2138.1 490 0
tt
Solving (positive roots only)
0.59451 mint and 1.52411 mint
PROBLEM 5.146
Determine the y coordinate of the centroid of the body shown.
SOLUTION
First note that the values of
Y
will be the same for the given body and the body shown below. Then
PROBLEM 5.147
An 8-in.-diameter cylindrical duct and a 4 8-in.
rectangular duct are to be joined as indicated.
Knowing that the ducts were fabricated from the
same sheet metal, which is of uniform thickness,
locate the center of gravity of the assembly.
SOLUTION
Assume that the body is homogeneous so that its center of gravity coincides with the centroid of the area.
By symmetry,
0.Z
PROBLEM 5.148
Three brass plates are brazed to a steel pipe to form the flagpole
base shown. Knowing that the pipe has a wall thickness of 8 mm
and that each plate is 6 mm thick, determine the location of the
center of gravity of the base. (Densities: brass 8470 kg/m3;
steel 7860 kg/m3.)
SOLUTION
Since brass plates are equally spaced, we note that
the center of gravity lies on the y-axis.
Thus, 0XZ