Basics of Fluid Mechanics, 2014a

3.3. MOMENT OF INERTIA 61
Example 3.5:
Calculate the rectangular moment of Inertia
for the rotation trough center in zz
axis (axis of rotation is out of the page).
Hint, construct a small element and build
longer build out of the small one. Using
this method calculate the entire rectangular.
Solution
2b
dy
y
2a
dx
r
x
Fig. -3.10. Rectangular Moment of
inertia.
The moment of inertia for a long element with a distance y shown in Figure 3.10 is
dI zz | dy
=
∫ a
−a
r 2
{ }} { (
y 2 + x 2) dy dx = 2 ( 3 ay 2 + a 3)
3
dy
(3.V.a)
The second integration ( no need to use (3.20), why?) is
Results in
Or
Example 3.6:
I zz =
∫ b
−b
I zz = a ( 2 ab 3 +2a 3 b )
3
2 ( 3 ay 2 + a 3)
dy
3
=
End Solution
4 ab
{}}{
A
( (2a) 2 +(2b) 2 )
12
(3.V.b)
(3.V.c)
Calculate the center of area and moment of inertia
for the parabola, y = αx 2 , depicted in Figure 3.11.
Hint, calculate the area first. Use this area to calculate
moment of inertia. There are several ways
to approach the calculation (different infinitesimal
area).
Solution
Fig. -3.11. Parabola for calculations
of moment of inertia.
For y = b the value of x = √ b/α. First the area inside the parabola calculated as
∫ √ b/α { }} {
A =2 (b − αξ 2 )dξ =
0
dA/2
2(3 α − 1)
3
( b
α)3
2