The document discusses the topic of moment of inertia. It will cover moment of inertia concepts including introduction, parallel axis theorem, perpendicular axis theorem, and moment of inertia calculations for rectangles, triangles, and circles. Examples are provided for calculating the moment of inertia of a rectangle about its centroidal axis and for a triangular section about its base and centroidal axis. The last example calculates the second moment of inertia of a circular lamina about its centroidal axis using the polar moment of inertia method.
2. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
3. Topic will be cover:
1. Moment of Inertia:
h) Formula of Moment of Inertia
i) Steps of problem solving
j) Example (1 to 8 With Solution)
k) Example (Home Work)
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
4. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
5. a)
Introduction
๏ It is defined as the algebraic sum of product of area and the
square of distance from the particular axis.
๏ It is denoted by ๐ผ ๐๐ or ๐ผ ๐๐
๏ It is given by I =Aโ2
๏ Unit - ๐๐4
๏ It is second moment of area which is measure of rรฉsistance
to bending & forms basic to strength of material.
๏ Mass MI is the rรฉsistance to rotation & forms basic to
dynamics to rigid bodies.
6. ๏ Consider a lamina of area A shown in fig.
๏ Let, this lamina is split up in to an infinite number of small
elements each of area da.
๏ Let, x1, x2, x3 ... are the distances of small elements from
OY axis.
๏ y1 , y2, y3 ... are the distances of small elements from OX
axis.
๏ Taking second moment of all the small elements about OX
axis,
8. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
9. b)
ParallelAxis
Theorem
๏ M.I of plane section about any axis is equal to MI of plane
section about the parallel axis parallel axis passing through
its CG plus the product of area and square of distance
between two axes.
๏ ๐ฐ ๐จ๐ฉ = ๐ฐ ๐ + a๐ ๐
Where,
๐ผ๐ด๐ต = M.I. of area about AB
๐ผ๐ = M.I. of area about C.G.
a = Area of the section
h = distance between c.g.
of section and axis AB.
10. Proof:
Condition 1: Two axis must be parallel.
Condition 2: One axis should pass through CG.
Condition 3: h is perpendicular distance between two axis.
๏ Y = distance of strip from CG of section.
11. ๏ M.I. of strip about CG of section,
๏ I = da.๐ฆ2
๏ M.I. of whole section about an axis passing through CG,
๏ ๐ผ๐ = ๐๐. ๐ฆ2
๏ M.I. of whole section about an axis AB,
๏ ๐ผ๐ด๐ต = ๐๐. โ + ๐ฆ 2
= ๐๐. (โ2 + ๐ฆ2 + 2โ๐ฆ)
= ๐๐. โ2 + ๐๐. ๐ฆ2+ ๐๐. 2โ๐ฆ
= aโ2+ ๐ผ๐ + 0
๏ ๐ผ๐ด๐ต = ๐ผ๐ + aโ2
12. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
13. c)
Perpendicular
AxisTheorem
๏ The moment of inertia of polar axis is equal to algebraic
sum of MI of two axis about which axis is perpendicular.
๏ ๐ฐ ๐๐ = ๐ฐ ๐๐ + ๐ฐ ๐๐
Proof:
Condition 1: Only for 2D Body (Laminar Plate)
Condition 2: X,Y & Z axis mutually perpendicular
Condition 3: X &Y axis should be in the plane of the body
Condition 4: Z should be perpendicular to the plane of body
2D Body
14. ๏ Consider a small area da on lamina plate.
๏ X andY are co-ordinate of P along OX and OY axis.
๏ OP = r, (distance between P and z axis)
๏ Now use Pythagoras theorem,
๏ ๐2 = ๐2 + ๐2
๏ M.I. of P about X โ X axis,
๏ ๐ผ ๐๐ = ๐๐. ๐ฆ2
๏ M.I. of P about Y โY axis,
๏ ๐ผ ๐๐ = ๐๐. ๐ฅ2
2D Body
๏ M.I. of P about Z โ Z axis,
๏ ๐ผzz = ๐๐. r2
15. ๏ M.I. of P about Z โ Z axis,
๏ ๐ผzz = ๐๐. r2 = da (๐2 + ๐2) = ๐๐. ๐ฅ2 + ๐๐. ๐ฆ2
๏ ๐ผzz = ๐ผ ๐๐ + ๐ผ ๐๐
2D Body
16. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
17. Example on Second Moment of Inertia
Example 1: Determine the second moment of area of a rectangle about an axis through the centroid.
Solution:
Given Data,
๏ Consider a rectangle of width โbโ and depth โdโ.
๏ Also consider a small strip of thickness โdyโ at distance โyโ from centroidal axis
๏ We know that second moment of area of a strip about xx axis = da.๐ฆ2
Where, da = area of small strip
๏ For M.I. of whole area, integrate between limits
โ๐
2
๐๐๐
๐
2
โด ๐ผ ๐๐ =
โ๐
2
๐
2
๐๐. ๐ฆ2
โด ๐ผ ๐๐ =
โ๐
2
๐
2
(๐. ๐ ๐ฆ). ๐ฆ2
18. Example on Second Moment of Inertia
Example 1: Determine the second moment of area of a rectangle about an axis through the centroid.
Solution:
Given Data,
โด ๐ผ ๐๐ =
โ๐
2
๐
2
๐๐. ๐ฆ2
โด ๐ผ ๐๐ =
โ๐
2
๐
2
(๐. ๐ ๐ฆ). ๐ฆ2
โด ๐ผ ๐๐ = ๐
โ๐
2
๐
2
(๐ ๐ฆ). ๐ฆ2
โด ๐ผ ๐๐ = ๐
๐ฆ3
3 โ๐
2
๐
2
โด ๐ผ ๐๐ = b
๐3
24
โ
โ๐3
24
=
๐.๐3
12
๐ผ ๐๐ =
๐.๐3
12
19. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
20. d)
Polar Moment
of Inertia
๏ Moment of inertia about an axis perpendicular to the plane
of an area is known as polar moment of inertia.
๏ It may be denoted as J or ๐ผ๐ง๐ง.
๏ Thus, the moment of inertia about an axis perpendicular to
the plane of the area at O in Fig. is called polar moment of
inertia at point O, and is given by,
๐ผ๐ง๐ง = ๐2
da
21. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
22. Example on Second Moment of Inertia
Example 2: Calculate moment of inertia of a triangular section with base b and height h, about base and about axis
passing through centroid.
Solution:
Given Data, ๏ Consider a triangular section as shown in figure,
๏ let, base = b height = h
๏ Consider a small strip PQ of thickness dx at distance x from A.
โด
๐๐
๐ต๐ถ
=
๐ฅ
โ
โด ๐๐ = ๐ต๐ถ.
๐ฅ
โ
๏ But, BC = base = b,
โด ๐๐ =
๐. ๐ฅ
โ
๏ Now area of strip PQ =
๐.๐ฅ
โ
. dx
23. Example on Second Moment of Inertia
Example 2: Calculate moment of inertia of a triangular section with base b and height h, about base and about axis
passing through centroid.
Solution:
Given Data, โด ๐. ๐ผ. ๐๐ ๐ ๐ก๐๐๐ ๐๐๐๐ข๐ก ๐๐๐ ๐ ๐ต๐ถ, = ๐ด๐๐๐ ๐ ๐ท๐๐ ๐ก๐๐๐๐ 2 =
๐.๐ฅ
โ
. dx โ โ ๐ฅ 2
๏ M.I. of whole triangle section may be found out by integrating above equation
between limits 0 to h.
โด ๐ผ ๐ต๐ถ = 0
โ ๐.๐ฅ
โ
โ โ ๐ฅ 2
. dx =
๐
โ 0
โ
๐ฅ โ2
+ ๐ฅ2
โ 2. โ. ๐ฅ . ๐๐ฅ
โด ๐ผ ๐ต๐ถ =
๐
โ 0
โ
๐ฅโ2
+ ๐ฅ3
โ 2. โ. ๐ฅ2
๐๐ฅ =
๐
โ
โ2
.
๐ฅ2
2
+
๐ฅ4
4
โ
2.โ.๐ฅ3
3 0
โ
โด ๐ผ ๐ต๐ถ =
๐
โ
โ4
2
+
โ4
4
โ
2.โ4
3
=
๐
โ
6.โ4+3.โ4โ8โ4
12
=
๐
โ
โ4
12
=
๐.โ3
12
________M.I. about
Base BC.
24. Example on Second Moment of Inertia
Example 2: Calculate moment of inertia of a triangular section with base b and height h, about base and about axis
passing through centroid.
Solution:
Given Data, ๏ M.I. about c.g. of section can be found out by parallel axis theorem.
๏ ๐ผ ๐ต๐ถ = ๐ผ ๐ฅ๐ฅ๐บ + ๐ด๐2
๏ ๐ผ ๐ฅ๐ฅ๐ = ๐ผ ๐ต๐ถ โ ๐ด๐2
=
๐โ3
12
โ
1
2
๐โ
โ
3
2
=
๐.โ3
12
โ
๐.โ3
18
=
3๐.โ3โ2๐.โ3
36
๏ ๐ผ ๐ฅ๐ฅ๐ =
๐.โ3
36
___________________ M.I. about c.g. and parallel to base.
25. Topic will be cover:
1. Moment of Inertia:
a) Introduction
b) Parallel Axis Theorem
c) Perpendicular Axis Theorem
d) Moment Of Inertia of Rectangle
e) Polar Moment of Inertia
f) Moment of Inertia of Triangle
g) Moment Of Inertia of Circle
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
26. Example on Second Moment of Inertia
Example 3: Find second moment of inertia of a circular lamina, about its centroidal axis.
Solution:
Given Data,
๏Using method of polar M.I.
๏Consider a circular lamina of centre โOโ and radius โrโ.
๏Consider an elementary ring of radius โxโ and thickness โdxโ.
๏Area of elementary ring : da = 2.ฯ.x.dx
๏M.I. of ring about z-z axis = Area X ๐ท๐๐ ๐ก๐๐๐๐ 2
= 2.ฯ.x.dx X ๐ฅ 2
= 2.ฯ. ๐3
dx
๏Now M.I. of whole section about centroidal axis can be
found out by integrating the above equation from limits โOโ
to โrโ.
27. Example on Second Moment of Inertia
Example 3: Find second moment of inertia of a circular lamina, about its centroidal axis.
Solution:
Given Data,
โด ๐ผ๐ง๐ง =
0
๐
2. ๐. ๐3
. ๐๐ฅ
๐ผ๐ง๐ง= 2.๐.
๐4
4 0
๐
= 2.ฯ.
๐4
4
=
๐.๐4
2
โด ๐ผ๐ง๐ง =
๐
๐
2
4
2
โด ๐ผ ๐๐ =
๐
32
๐4
28. Example on Second Moment of Inertia
Example 3: Find second moment of inertia of a circular lamina, about its centroidal axis.
Solution:
Given Data,
๏ As per perpendicular axis theorem, ๐ผ ๐๐ = ๐ผ ๐๐
๏ Now, ๐ผ ๐๐ = ๐ผ ๐๐ + ๐ผ ๐๐ = 2 ๐ผ ๐๐
โด
๐
32
๐4 = 2 ๐ผ ๐๐
โด ๐ผ ๐๐ =
๐
64
๐4
โด ๐๐๐๐๐๐๐๐๐ฆ, ๐ผ ๐ฆ๐ฆ =
๐
64
๐4
29. Topic will be cover:
1. Moment of Inertia:
h) Formula of Moment of Inertia
i) Steps of problem solving
j) Example (1 to 8 With Solution)
k) Example (Home Work)
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
30. Formula of Moment of Inertia
Sr. No. Geometrical Shape Area ๐ฐ ๐๐ ๐ฐ ๐๐
1
b X d ๐ผ ๐ฅ๐ฅ =
๐๐3
12
๐ผ ๐ฆ๐ฆ =
๐๐3
12
2
๐
4
๐2 ๐ผ ๐ฅ๐ฅ =
๐
64
๐4
๐ผ ๐ฆ๐ฆ =
๐
64
๐4
31. Formula of Moment of Inertia
Sr. No. Geometrical Shape Area ๐ฐ ๐๐ ๐ฐ ๐๐
3
1
2
๐.h
๐ผ ๐ฅ๐ฅ =
๐. โ3
36
๐ผ ๐ฆ๐ฆ =
โ. ๐3
48
4
1
2
๐. h ๐ผ ๐ฅ๐ฅ =
๐. โ3
36
๐ผ ๐ฆ๐ฆ =
โ. ๐3
36
32. Topic will be cover:
1. Moment of Inertia:
h) Formula of Moment of Inertia
i) Steps of problem solving
j) Example (1 to 8 With Solution)
k) Example (Home Work)
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
33. h)
Steps of
solving
problem of
M.I.
Number Discription
Step 1: Split figure from unknown geometry to known geometry and draw X
andY Axis.
Like โ Square, Rectangle,Triangle, Circle etc.
Step 2: Find Centroid of Individual geometry (X andY Component)
Like - ๐1, ๐1, ๐2, ๐2 (Use Centroid Formula)
Step 3: Find centroid of whole geometry.
Like - ๐ & ๐
Step 4: Find Moment about x axis of Individual geometry
(i.e.: ๐ผ ๐๐1, ๐ผ ๐๐2, ๐ผ ๐๐3, ๐ผ ๐๐4)
(Use parallel axis theorem, ๐ผ ๐๐1 = ๐ผ๐ + ๐โ2 )
Where, ๐ผ๐ = ๐๐๐๐๐๐ก ๐๐ ๐๐๐๐๐ก๐๐ ๐๐ ๐๐๐๐๐ฃ๐๐๐ข๐๐ ๐๐๐๐๐๐ก๐๐ฆ ๐ผ ๐๐
a = Area of geometry
h = ๐1 - ๐
Step 5: Find Moment aboutY axis of Individual geometry
(i.e.: ๐ผ ๐๐1, ๐ผ ๐๐2, ๐ผ ๐๐3, ๐ผ ๐๐4)
(Use parallel axis theorem, ๐ผ ๐๐1 = ๐ผ๐ + ๐โ2
)
Where, ๐ผ๐ = ๐๐๐๐๐๐ก ๐๐ ๐๐๐๐๐ก๐๐ ๐๐ ๐๐๐๐๐ฃ๐๐๐ข๐๐ ๐๐๐๐๐๐ก๐๐ฆ ๐ผ ๐๐
a = Area of geometry
h = ๐1 - ๐
34. Number Discription
Step 6: Find Total M.I. of geometry (X andY Component)
Like - ๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3 + ๐ผ ๐๐4 + ________
Like - ๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3 + ๐ผ ๐๐4 + ________
35. Topic will be cover:
1. Moment of Inertia:
h) Formula of Moment of Inertia
i) Steps of problem solving
j) Example (1 to 8 With Solution)
k) Example (Home Work)
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
36. Example on Moment of Inertia
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
Step 1:Split figure from unknown geometry to known
geometry and draw X andY Axis.
20 cm
2 cm
30 cm
2 cm
37. Example on Composite Area (2D)
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
Now Consider Element 1
Area of Element 1 = 2 x 20 = 40 ๐ถ๐2
๐1=
๐ฟ
2
=
20
2
= 10 cm
Y1 = 30 +
๐
2
= 30 +
2
2
= 30 + 1 = 31 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
X1
Y1
Step 2 :Find Centroid of Individual geometry (X andY Component)
Like - ๐ฟ ๐, ๐ ๐, ๐ฟ ๐, ๐ ๐ (Use Centroid Formula)
38. Example on Composite Area (2D)
20 cm
2 cm
30 cm
2 cm
Now Consider Element 2
Area of Element 2 = 2 x 30 = 60 ๐๐2
๐2= 10 cm
Y2 =
๐
2
=
30
2
= 15 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
X2 Y2
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
39. Example on Composite Area (2D)
20 cm
2 cm
30 cm
2 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
Element Area (๐๐ ๐ ) Xi Yi
1 40 X1=10 cm Y1=31 cm
2 60 X2=10 cm Y2=15 cm
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2
๐ด1+๐ด2
=
400+600
100
= 10 cm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2
๐ด1+๐ด2
=
1240+900
100
= 21.40 cm
G
Step 3: Find centroid of whole geometry.
Like - ๐ฟ & ๐
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
40. Example on Composite Area (2D)
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
๐ฟ
๐
G
Step 4: Find Moment about x axis of Individual geometry
Like - ๐ฐ ๐ฟ๐ฟ๐, ๐ฐ ๐ฟ๐ฟ๐
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐1 =
๐๐3
12
=
(20)(2)3
12
= 13.33 ๐๐4
Area, a = b X d = 20 X 2 = 40 ๐๐2
h = ๐1 - ๐ = 31 โ 21.40 = 9.6 cm
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 13.33 + (40 X 9.62)
= 3699.73 ๐๐4
X1
Y1h
41. Example on Composite Area (2D)
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
๐
G
Step 4: Find Moment about x axis of Individual geometry
Like - ๐ฐ ๐ฟ๐ฟ๐, ๐ฐ ๐ฟ๐ฟ๐
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐2 =
๐๐3
12
=
(2)(30)3
12
= 4500 ๐๐4
Area, a = b X d = 2 X 30 = 60 ๐๐2
h = ๐2 - ๐ = 15 โ 21.40 = (-6.4) cm
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 4500 + (60 X 6.42)
= 6957.6 ๐๐4
h
X2 Y2
42. Example on Composite Area (2D)
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
๐ฟ
๐
G
Step 5: Find Moment aboutY axis of Individual geometry
Like - ๐ฐ ๐๐๐, ๐ฐ ๐๐๐
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐1 =
๐๐3
12
=
(2)(20)3
12
= 1333.33 ๐๐4
Area, a = b X d = 20 X 2 = 40 ๐๐2
h = ๐1 - ๐ = 10 โ 10 = 0 cm
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 1333.33 + (40 X0)
= 1333.33 ๐๐4
X1
Y1
43. Example on Composite Area (2D)
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
๐
G
Step 5: Find Moment about x axis of Individual geometry
Like - ๐ฐ ๐๐๐, ๐ฐ ๐๐๐
๐ฐ ๐๐๐ = ๐ฐ ๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐2 =
๐๐3
12
=
(30)(2)3
12
= 20 ๐๐4
Area, a = b X d = 2 X 30 = 60 ๐๐2
h = ๐2 - ๐ = 0 โ 0 = 0 cm
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 20 + (60 X0)
= 20 ๐๐4
X2 Y2
๐ฟ
44. Example on Composite Area (2D)
Example 1: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a โTโ section as per figure.
Solution:
Given Data,
20 cm
2 cm
30 cm
2 cm
๐
G
Step 6: Find Total M.I. of geometry (X andY Component)
Like - ๐ฐ ๐ฟ๐ฟ, ๐ฐ ๐๐
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2
๐ผ ๐๐ = 3699.7 + 6957.6
๐ผ ๐๐ = 10657.3 ๐๐4
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2
๐ผ ๐๐ = 1333.33 + 20
๐ผ ๐๐ = 1353.33 ๐๐4
X2
๐ฟ
45. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
2.5
cm
10 cm
2.5 cm
10 cm
2.5 cm
2.5
cm
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data, Step 1:Split figure from unknown geometry to known
geometry and draw X andY Axis.
46. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
2.5
cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
Now Consider Part 1
Area of Element 1 = 2.5 x 10 = 25 ๐๐2
๐1=
๐ฟ
2
=
10
2
= 5 cm
Y1 = 2.5 + 10 +
๐
2
= 2.5 + 10 +
2.5
2
= 13.75 cm
X1
Y1
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data, Step 2 :Find Centroid of Individual geometry (X andY Component)
Like - ๐ฟ ๐, ๐ ๐, ๐ฟ ๐, ๐ ๐ (Use Centroid Formula)
47. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
Now Consider Part 2
Area of Element 1 = 2.5 x 10 = 25 ๐๐2
๐2=
๐ฟ
2
=
2.5
2
= 1.25 cm
Y2 = 2.5 +
๐
2
= 2.5 +
10
2
= 7.5 cm
X2
Y2
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
48. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
2.5
cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
Now Consider part 3
Area of Element 3 = 2.5 x 10 = 25 ๐๐2
๐3=
๐ฟ
2
=
10
2
= 5 cm
Y3 =
๐
2
=
2.5
2
= 1.25 cm
X3
Y3
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
49. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
Part Area (๐๐ ๐ ) Xi Yi
1 25 X1=5 cm Y1=13.75 cm
2 25 X2=1.25 cm Y2=7.5 cm
3 25 X3=5 cm Y3=1.25 cm
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2+๐ด3 ๐3
๐ด1+๐ด2+๐ด3
=
125+31.25+125
75
= 3.75 cm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2+๐ด3 ๐3
๐ด1+๐ด2+๐ด3
=
343.75+187.5+31.25
75
= 7.5 cm
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
Step 3: Find centroid of whole geometry.
Like - ๐ฟ & ๐
50. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
Step 4: Find Moment about x axis of Individual geometry
Like - ๐ฐ ๐ฟ๐ฟ๐, ๐ฐ ๐ฟ๐ฟ๐
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐1 =
๐๐3
12
=
(10)(2.5)3
12
= 13.02 ๐๐4
Area, a = b X d = 10 X 2.5 = 25 ๐๐2
h = ๐1 - ๐ = 13.75 โ 7.5 = 6.25 cm
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 13.02 + (25 X 6.252)
= 989.58 ๐๐4
X1
Y1
51. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐2 =
๐๐3
12
=
(2.5)(10)3
12
= 208.33 ๐๐4
Area, a = b X d = 10 X 2.5 = 25 ๐๐2
h = ๐2 - ๐ = 7.5 โ 7.5 = 0 cm
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 208.33 + (25 X0)
= 208.33 ๐๐4
X2
Y2
52. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐3 =
๐๐3
12
=
(10)(2.5)3
12
= 13.02 ๐๐4
Area, a = b X d = 2.5 X 10 = 25 ๐๐2
h = ๐3 - ๐ = 1.25 โ 7.5 = - 6.25 cm
๐ฐ ๐ฟ๐ฟ๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 13.02 + (25 X6.252)
= 989.58 ๐๐4
X3
Y3
53. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
Step 5: Find Moment aboutY axis of Individual geometry
Like - ๐ฐ ๐๐๐, ๐ฐ ๐๐๐
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐1 =
๐๐3
12
=
(2.5)(10)3
12
= 208.33 ๐๐4
Area, a = b X d = 10 X 2.5 = 25 ๐๐2
h = ๐1 - ๐ = 5 โ 3.75 = 1.25 cm
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 208.33 + (25 X 1.252)
= 247.39 ๐๐4
X1
Y1
54. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐2 =
๐๐3
12
=
(10)(2.5)3
12
= 13.02 ๐๐4
Area, a = b X d = 10 X 2.5 = 25 ๐๐2
h = ๐2 - ๐ = 1.25 โ 3.75 = -2.5 cm
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 13.02 + (25 X 2.52)
= 169.27 ๐๐4
X2
Y2
55. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐
๐ผ๐ = M.I. of area about C.G.
Letโs Find ๐ผ๐3 =
๐๐3
12
=
(2.5)(10)3
12
= 208.33 ๐๐4
Area, a = b X d = 10 X 2.5 = 25 ๐๐2
h = ๐3 - ๐ = 5 โ 3.75 = 1.25 cm
๐ฐ ๐๐๐ = ๐ฐ ๐๐ + ๐๐ ๐ = 208.33 + (25 X 1.252)
= 247.39 ๐๐4
X3
Y3
56. Example on Composite Area (2D)
10 cm
2.5 cm
10 cm
2.5 cm
Note:
๐ = Distance between Centroid of element toY axis,
๐ = Distance between Centroid of element to X axis,
๐ฟ
๐
G
Example 2: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for as per figure.
Solution:
Given Data,
Step 6: Find Total M.I. of geometry (X andY Component)
Like - ๐ฐ ๐ฟ๐ฟ, ๐ฐ ๐๐
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3
๐ผ ๐๐ = 989.58 + 208.33 + 989.58
๐ผ ๐๐ = 2187.49 ๐๐4
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3
๐ผ ๐๐ = 247.39 + 169.27 +247.39
๐ผ ๐๐ = 664.05 ๐๐4
57. Example on Composite Area (2D)
Example 3: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
2 cm
2 cm
2 cm
8 cm
16 cm
Part Area (๐๐ ๐ ) Xi Yi
1 16 X1=4 cm Y1=13 cm
2 20 X2=7 cm Y2=7 cm
3 32 X3=14 cm Y3=1 cm
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2+๐ด3 ๐3
๐ด1+๐ด2+๐ด3
=
64+140+448
68
= 9.58 cm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2+๐ด3 ๐3
๐ด1+๐ด2+๐ด3
=
208+140+32
68
= 5.58 cm
๐ฟ ๐
G
12 cm
Y1=13 cm,Y2=7 cm,Y3=1 cm, ๐ = 5.58 cm
58. Example on Composite Area (2D)
Example 3: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
2 cm
2 cm
2 cm
8 cm
16 cm
๐ฟ ๐
12 cm
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3
๐ผ ๐๐ = 886.23+206.99+681.99
๐ผ ๐๐ = 1775.21 ๐๐4
1
2
3
๐. ๐3
12
= 5.33
๐. ๐3
12
= 166.67
๐. ๐3
12
= 10.66
16
20
32
7.42
1.42
-4.58
๐ฐ ๐ฟ๐ฟ๐ = 886.23
๐ฐ ๐ฟ๐ฟ๐ = 206.99
๐ฐ ๐ฟ๐ฟ๐ = 681.90
Y1=13 cm,Y2=7 cm,Y3=1 cm, ๐ = 5.58 cm
59. Example on Composite Area (2D)
Example 3: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
2 cm
2 cm
2 cm
8 cm
16 cm
๐ฟ ๐
G
12 cm
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3
๐ผ ๐๐ = 583.51+139.80+1307.83
๐ผ ๐๐ = 2031.14 ๐๐4
1
2
3
๐. ๐3
12
= 85.33
๐. ๐3
12
= 6.67
๐. ๐3
12
= 682.67
16
20
32
-5.58
-2.58
4.42
๐ฐ ๐๐๐ = 583.51
๐ฐ ๐๐๐ = 139.80
๐ฐ ๐๐๐ = 1307.83
X1=4 cm, X2=7 cm, X3=14 cm, ๐ฟ = 9.58 cm
60. Example on Composite Area (2D)
Example 4: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
9cm
3 cm
6 cm
๐ฟ
๐
Element Area (๐๐ ๐ ) Xi Yi
1 27 X1=1.5 cm Y1=4.5 cm
2 13.5 X2=4 cm Y2=3 cm
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2
๐ด1+๐ด2
= 2.33 cm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2
๐ด1+๐ด2
= 4 cm
61. Example on Composite Area (2D)
Example 4: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
9cm
3 cm
6 cm
๐ฟ
๐
1
2
๐. ๐3
12
= 182.25
๐.โ3
36
= 60.75
27
13.5
Y1=4.5 cm,Y2=3 cm, ๐ = 4 cm
0.5
- 1
189
74.25
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2
๐ผ ๐๐ = 189+74.25
๐ผ ๐๐ = 263.25 ๐๐4
62. Example on Composite Area (2D)
Example 4: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
9cm
3 cm
6 cm
๐ฟ
๐
1
2
๐. ๐3
12
= 20.25
โ.๐3
36
= 6.75
27
13.5
-0.83
1.67
38.85
44.40
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2
๐ผ ๐๐ = 38.85+44.40
๐ผ ๐๐ = 83.25 ๐๐4
X1=1.5 cm, X2=4 cm, ๐ฟ = 2.33 cm
63. Example on Composite Area (2D)
Example 5: A rectangular section is 400 mm wide and 800 mm deep. Two circular holes of 200 mm diameter each are
cut on Y-Y axis at distance 200 mm and 600 mm from the top edge. Calculate M.I. about xx and yy axis.
Solution:
Given Data,
800
400
200200200200
1
2
3
A1 = 320000
A2 = 31415.92
A3=31415.92
X1= 0
X2= 0
X3= 0
Y1= 0 mm
Y2= 200 mm
Y3= -200 mm
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1โ ๐ด2 ๐2โ๐ด3 ๐3
๐ด1+๐ด2+๐ด3
= 0 mm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1โ ๐ด2 ๐2โ๐ด3 ๐3
๐ด1โ๐ด2โ๐ด3
= 0 mm
64. Example on Composite Area (2D)
Example 5: A rectangular section is 400 mm wide and 800 mm deep. Two circular holes of 200 mm diameter each are
cut on Y-Y axis at distance 200 mm and 600 mm from the top edge. Calculate M.I. about xx and yy axis.
Solution:
Given Data,
800
400
200200200200
1
2
3
๐
64
๐4 = 7.85๐107
๐
64
๐4 = 7.85๐107
๐. ๐3
12
= 1.7๐1010 320000
31415.92
31415.92
0
200
200
1.7๐1010
1.3๐109
1.3๐109
๐ผ ๐๐ = ๐ผ ๐๐1 โ ๐ผ ๐๐2 โ ๐ผ ๐๐3
๐ผ ๐๐ = (1.7๐1010) โ 1.3๐109 โ 1.3๐109
๐ผ ๐๐ = 1.44 ๐ 1010 ๐๐4
Y1=0 mm,Y2=200 mm,Y3=-200 mm, ๐ = 0 mm
65. Example on Composite Area (2D)
Example 5: A rectangular section is 400 mm wide and 800 mm deep. Two circular holes of 200 mm diameter each are
cut on Y-Y axis at distance 200 mm and 600 mm from the top edge. Calculate M.I. about xx and yy axis.
Solution:
Given Data,
800
400
200200200200
1
2
3
๐. ๐3
12
= 4.27๐109
๐
64
๐4
= 7.85๐107
๐
64
๐4 = 7.85๐107
320000
31415.92
31415.92
0
0
0
4.27๐109
7.85๐107
7.85๐107
๐ผ ๐๐ = ๐ผ ๐๐1 โ ๐ผ ๐๐2 โ ๐ผ ๐๐3
IYY = (4.27X109) โ(7.85X107) โ (7.85X107)
๐ผ ๐๐ = 4.1 ๐109 ๐๐4
X1=0 mm, X2=0 mm, X3=0 mm, ๐ฟ = 0 mm
66. Example 6: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
Example on Composite Area (2D)
1
2
3
A1 = 30X100=3000
A2 = 25X100=2500
X1= 100 Y1= 20+20+80+
30
2
= 135
Y2= 20+
100
2
= 70
Y4= 20+
20
3
= 26.664
5 A5 =
1
2
๐87.5๐20=875
A4 =
1
2
๐87.5๐20=875
A3 = 20X200=4000
X2= 100
X3= 100
X4=
2
3
๐87.5 = 58.33
X5=
87.5+25+
87.5
3
=141.67
Y3=
20
2
= 10
Y5= 20+
20
3
= 26.66
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2+๐ด3 ๐3+๐ด4 ๐4+๐ด5 ๐5
๐ด1+๐ด2+๐ด3+๐ด4+๐ด5
= 100 mm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2+๐ด3 ๐3+๐ด4 ๐4+๐ด5 ๐5
๐ด1+๐ด2+๐ด3+๐ด4+๐ด5
= 59.26 mm
67. Example 6: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
Example on Composite Area (2D)
1
2
3
4
5
๐. ๐3
12
= 2.25๐105
๐. ๐3
12
= 2.08๐106
๐. ๐3
12
= 1.33๐105
๐. โ3
36
= 1.94๐104
๐. โ3
36
= 1.94๐104
3000
2500
4000
875
875
Y1=135,Y2=70,Y3=10,Y4=26.66,Y5=26.66, ๐ = 69.91 mm
65.09
0.09
-59.91
-43.25
-43.25
1.29๐107
2.08๐106
1.45๐107
1.66๐106
1.66๐106
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3 + ๐ผ ๐๐4 + ๐ผ ๐๐5
๐ฐ ๐ฟ๐ฟ = 3.28 X ๐๐ ๐ ๐๐ ๐
68. Example 6: Calculate ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ for a given figure.
Solution:
Given Data,
Example on Composite Area (2D)
1
2
3
4
5
๐. ๐3
12
= 2.5๐106
๐. ๐3
12
= 1.30๐105
๐. ๐3
12
= 1.33๐107
โ.๐3
36
= 3.72๐105
โ.๐3
36
= 3.72๐105
3000
2500
4000
875
875
X1=X2= X3=100, X4=58.33, X5=141.67, ๐ฟ = 100 mm
0
0
0
-41.67
41.67
2.5๐106
1.30๐105
1.33๐107
1.89๐106
1.89๐106
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 + ๐ผ ๐๐3 + ๐ผ ๐๐4 + ๐ผ ๐๐5
๐ฐ ๐๐ = 1.97 X ๐๐ ๐ ๐๐ ๐
69. Example on Composite Area (2D)
Example 7: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data, Part Area (๐๐ ๐ ) Xi (mm) Yi (mm)
1 1,20,000 X1=200 Y1= 150
2 30,000 X2=400+
200
3
= 466.67 Y2=
300
3
= 100
3 ๐.๐2
2
=35342.91 X3= 100 + 150 =250 Y3=
4.๐
3.๐
= 63.66
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2โ๐ด3 ๐3
๐ด1+๐ด2โ๐ด3
= 254.36 mm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2โ๐ด3 ๐3
๐ด1+๐ด2โ๐ด3
= 163.53 mm
100 300 200
300
1
2
3
70. Example on Composite Area (2D)
Example 7: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data,
100 300 200
300
1
2
3
1
2
3
๐. ๐3
12
= 9๐108
0.11.๐4
= 5.57๐107
๐. โ3
36
= 1.5๐108
1,20,000
30,000
35342.91
Y1=150,Y2=100,Y3=63.66, ๐ = 163.53 mm -13.53
-63.53
-99.87
9.21๐108
2.71๐108
4.08๐108
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 โ ๐ผ ๐๐3
๐ฐ ๐ฟ๐ฟ = 7.84 X ๐๐ ๐
๐๐ ๐
71. Example on Composite Area (2D)
Example 7: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data,
100 300 200
300
1
2
3
1
2
3
๐. ๐3
12
= 1.6๐109
โ.๐3
36
= 6.67๐107
๐.๐4
8
= 1.99๐108
1,20,000
30,000
35342.91
X1=200, X2=466.67, X3=250, ๐ฟ = 254.36 mm
-54.36
212.31
-4.36
1.95๐109
1.41๐109
1.99๐108
๐ผ ๐๐ = ๐ผ ๐๐1 + ๐ผ ๐๐2 โ ๐ผ ๐๐3
๐ฐ ๐๐ = 3.16 X ๐๐ ๐
๐๐ ๐
72. Example on Composite Area (2D)
Example 8: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data, Part Area
(๐๐ ๐
)
Xi (mm) Yi (mm)
1 12X9=108 X1=
12
2
= 6 Y1=
9
2
= 4.5
2 27 X2= 6+
2
3
(6) = 10 Y2=
1
3
(9) = 3
3 9 X3=
1
3
(6) = 2 Y3=6+
2
3
(3) = 8
6 6
66
6
3
1 2
3
๐ = Distance between Centroid of element toY axis,
๐ =
๐ด1 ๐1โ ๐ด2 ๐2โ๐ด3 ๐3
๐ด1โ๐ด2โ๐ด3
= 5 mm
๐ = Distance between Centroid of element to X axis,
๐ =
๐ด1 ๐1+ ๐ด2 ๐2โ๐ด3 ๐3
๐ด1+๐ด2โ๐ด3
= 4.63 mm
73. Example on Composite Area (2D)
Example 8: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data,
6 6
66
6
3
1 2
3
1
2
3
๐. ๐3
12
= 729
๐. โ3
36
= 121.5
๐. โ3
36
= 4.5
108
27
9
Y1=4.5,Y2=3,Y3=8, ๐ = 4.63 mm -0.13
-1.63
3.37
730.82
193.24
106.71
๐ผ ๐๐ = ๐ผ ๐๐1 โ ๐ผ ๐๐2 โ ๐ผ ๐๐3
๐ฐ ๐ฟ๐ฟ = ๐๐๐. ๐๐ ๐๐ ๐
74. Example on Composite Area (2D)
Example 8: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data,
6 6
66
6
3
1 2
3
1
2
3
108
27
9
1
5
-3
1404
729
99
๐ผ ๐๐ = ๐ผ ๐๐1 โ ๐ผ ๐๐2 โ ๐ผ ๐๐3
๐ฐ ๐๐ = ๐๐๐ ๐๐ ๐
๐. ๐3
12
= 1296
โ.๐3
36
= 54
โ.๐3
36
= 18
X1=6, X2=10, X3=2, ๐ฟ = 5 mm
75. Topic will be cover:
1. Moment of Inertia:
h) Formula of Moment of Inertia
i) Steps of problem solving
j) Example (1 to 8 With Solution)
k) Example (Home Work)
Department of Mechanical
Engineering.
Prof. Malay Badodariya
malay.badodariya@marwadie
ducation.edu.in
Unit no.: 5
Subject Name: FMD
01ME0305
76. Example on Composite Area (2D)
Example 9: Determine location of centroid, ๐ฐ ๐ฟ๐ฟ ๐๐๐ ๐ฐ ๐๐ of a lamina shown in figure. (All Dimension in mm)
Solution:
Given Data,
3
6
3
3 6
3
3
6
4