MEEG 112, Statics: Spring 2010
1.
"
"
a) Determine the centroid ( x , y ) of the shaded area, then:
b) Calculate the moment of inertia of the shaded area about the y-axis and, then,
c) Calculate its moment of inertia about an axis parallel to the y-axis but passing
"
through the centroid y . !
!
x-
⌠
⌡ x dA
& y- =
= ⌠ dA
⌡
⌠
⌡ y dA
etc.
⌠ dA
⌡
Calculate ⌠
⌡ dA first using a horizontal elemental slice of area:
2
1/2 dy = [2. y3/2]1
=⌠
⌡ x dy = ⌠
⌡ 2y
0
3
a)
Calculate
∴
4
2
⌠
⌡ dA = 3 in .
x- using a horizontal elemental slice of area:
⌠
⌠
⌡ x dA
⌡ x dA
= ⌠ dA
=
x
= 4/3
⌡
x
x
.
⌠
2 dy
⌡0
1
=
Calculate
(4/3)
=
⌠
⌡ 2y dy
4/3
∴
3
x- = 4 in.
y- using a horizontal elemental slice of area:
⌠ y dA
⌠ y dA
⌠ y x dy
⌡
⌡
⌡
= ⌠ dA
=
=
4/3
4/3
⌡
3/2
⌠
⌡ 2y dy
=
4/3
∴
3
y- = 5 in
b) Moment of Inertia about y-axis. Use vertical slice because all the elemental area is at
the same distance from y-axis.
Iy
x2
2 dA = ⌠ x2 (1-y) dx = ⌠ x2 (1= ⌠
x
) dx
⌡
⌡
⌡
4
2
= ⌠
⌡ x -
x2
dx
4
=
[
x3 x5 2
]
3 5 0
∴
Iy = 1.067 in4
If you use a horizontal slice, all portions are at different distances from the y-axis so you
must find M. of I. about the centroid and then use the Parallel Axis Theorem to shift to
the y-axis.
hb3
b3 dy
x3 dy
x 2
2A = ⌠
Iy
=
+ d2 A
= ⌠
+
d
+ ⌠
( ) x dy
⌡
⌡
⌡
12
12
12
2
8 y3/2 dy
= ⌠
+ ⌠1 (2y3/2) dy ∴
Iy = 1.067 in4
⌡ 12
⌡0
c) Moment of Inertia about centroidal axis parallel to y-axis.
Iy
= I y + A d2
4 3
1.067 = I y + 3 (4 ) 2
Iy =0.317 in4
2.
"
"
Locate the centroid ( x , y ) of the cross-sectional area of the channel.
!
3.
Determine the smallest horizontal force P required just to move the block A to the
right if the spring force is 600 N and the coefficient of static friction at all contacting
surfaces on A is µs = 0.3. The inside of the sleeve C is smooth. Neglect the masses of A
and B.
4.
The engine hoist is used to support the 200 kg engine. Determine the force acting
in the hydraulic cylinder AB, the horizontal and vertical components of force at the pin C
and the reactions at the fixed support D.
5.
Determine the force in each member of the truss and state if the members are in
tension or compression. The load has a mass of 40 kg.
(By inspection members EC, FC, FB, BG are all immediately zero force members)
6.
Determine the resultant moment produced by forces FB and FC about point O.
Express the result as a Cartesian vector.