PUSAT MASSA INTEGRAL

1. Mathematics II

2. Center of Mass

3. Definition
• Mass is a measure for the amount of substance / amount of substance
contained in an object
• Weight is the mass of an object that is affected by the force of gravity
• The center of mass and center of gravity is a point where the center of
the mass or weight of a line (1-D object), a plane (2-D object ) or a
space (3-D object) is concentrated. The location of the center of mass
is not affected by the gravity, while the center of gravity is influenced
by the gravity. Thus, the location of the center of mass does not always
coincide with the location of the center of gravity.
• Location of the center of mass:
– Located in the middle of a straight (homogeneous) line
– Located at the intersection of the diagonal of a plane (2-D object) and space (3-D
object0 for a homogeneous object in a regular shape.
– It can be located inside or outside the object depending on its homogeneity and
shape.

4. • Length is the measure of the line (1-D object) occupied by a substance
• Area is the size of the plane (2-D object) occupied by a substance
• Volume is a measure of the space (3-D object) occupied by a
substance
• Mass density is a measure of the concentration of mass on an object.
If the density is the same throughout the object, it is called
homogeneous or has a constant density.
Dimension Mass Density Formula Unit (IS)
1 l l = M/L kg/m
2 s s = M/A kg/m2
3 r r = M/V kg/m3

5. Center of Mass of a Plane
0 X-axis
Y-axis
• The picture beside is a plane of mass M
• The plane is divided (vertically) into n partitions of
mass m M
mi
• See the partition i of mass mi.
• The partition has a center of mass in the middle of
plane at the coordinate (xi
*, yi
*)
yi*
xi*
• Each partition m has center of mass (xi
*, yi
*); i =
1,2, .. n
• On the whole, the object of mass M has a center of
mass at the coordinate ( ҧ
𝑥, ത
𝑦)
0 X-axis
Y-axis
ҧ
𝑥
ത
𝑦
• To find the center of mass
ҧ
𝑥, ത
𝑦 , we will use the concept of first moment

6. Concept of First Moment
• The first moment (ML) of a plane relative to a line L
is the product of multiplication of the mass and the
direct distance between the center of mass and the
line L.
0 X-axis
Y-axis
ҧ
𝑥
ത
𝑦
• The moment on the Y-axis is called My, and the
moment on the X-axis is called Mx
𝑀𝑦 = ҧ
𝑥. 𝑀
𝑀𝑥 = ത
𝑦. 𝑀

7. 0 X-axis
Y-axis
• M, My, and Mx actually are the limit of summation
of the mass, the Y-moment and the X-moment
from all partition (see the picture in the right)
M
mi
• For the partition i of mass mi, with center of
mass (xi*, yi*) :
yi*
xi*
• For all partitions of the plane, then :
∆myi = xi* . ∆mi
∆mxi = yi* . ∆mi
𝑀𝑦 ≈ 𝑥1
∗
. ∆𝑚1 + 𝑥2
∗
. ∆𝑚2 + ⋯ + 𝑥𝑖
∗
. ∆𝑚𝑖 + ⋯ + 𝑥𝑛
∗. ∆𝑚𝑛
𝑀 ≈ ∆𝑚1 + ∆𝑚2 + ⋯ + ∆𝑚𝑖 + ⋯ + ∆𝑚𝑛
≈ ෍
𝑖=1
𝑛
∆𝑚𝑖
≈ ෍
𝑖=1
𝑛
𝑥𝑖
∗
. ∆𝑚𝑖
𝑀𝑥 ≈ 𝑦1
∗
. ∆𝑚1 + 𝑦2
∗
. ∆𝑚2 + ⋯ + 𝑦𝑖
∗
. ∆𝑚𝑖 + ⋯ + 𝑦𝑛
∗. ∆𝑚𝑛 ≈ ෍
𝑖=1
𝑛
𝑦𝑖
∗
. ∆𝑚𝑖

8. Thus, for n approaching infinity, the limits are :
Since M = σ A,
then dM = σ dA
My
dM
Mx dM
(σ = mass density for 2-D objects)
So, center of mass of a plane (for homogenous s) can be expressed :

9. Center of mass of plane under a curve
X-axis
a=0
Y-axis
)
(x
f
y =
b
xi
)
( i
x
f
Ai
Xi
*
Ai have a mass of mi.
mi = s Ai = s f(xi) xi
Accumulate Ai and mi, take the
limit, then state in integral form:
Ai = f(xi) xi
𝐴 = න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥
𝑀 = σ න
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = σ. 𝐴
Note : in this case, xi* is located in the middle of partition i

10. X-axis
a=0
Y-axis
)
(x
f
y =
b
xi
yi
*
The partition i has center of mass (xi*, yi*)
Ai
xi
*
Identify the position of xi* and yi*
xi* = xi and
𝑀𝑦 = 𝜎 න
𝑎
𝑏
𝑥 𝑑𝐴 = 𝜎 න
𝑎
𝑏
𝑥. 𝑓 𝑥 𝑑𝑥
yi* = ½ f(xi)
∆𝑚𝑦𝑖= 𝑥𝑖
∗
∆𝑚𝑖 = 𝑥𝑖
∗
𝜎∆𝐴𝑖
Identify the moments of ∆myi and ∆mxi
∆𝑚𝑥𝑖= 𝑦𝑖
∗
∆𝑚𝑖 = 𝑦𝑖
∗
𝜎∆𝐴𝑖
Accumulate myi and mxi, take the limit,
then state in integral form:
𝑀𝑥 = 𝜎 න
𝑎
𝑏
𝑦 𝑑𝐴 = 𝜎 න
𝑎
𝑏
1
2
𝑓 𝑥 . 𝑓 𝑥 𝑑𝑥 =
𝜎
2
න
𝑎
𝑏
𝑓 𝑥 2
𝑑𝑥
So, the center of mass of
the plane will be:
=
𝜎 ‫׬‬𝑎
𝑏
𝑥. 𝑓 𝑥 𝑑𝑥
𝜎𝐴
=
1
2
𝜎 ‫׬‬
𝑎
𝑏
𝑓 𝑥 2
𝑑𝑥
𝜎𝐴

11. sb. y
b
a
)
(x
f
y =
)
(x
g
y =
0 sb. x
xi
xi
)
(
)
( i
i x
g
x
f −
Ai
Note : in this case, xi* is located in the middle of partition i
Ai have a mass of mi.
mi = s Ai = s (f(xi)-g(xi)) xi
Accumulate Ai and mi, take the
limit, then state in integral form:
Ai = f(xi)-g(xi) xi
𝐴 = න
𝑎
𝑏
(𝑓 𝑥 − 𝑔 𝑥 )𝑑𝑥
𝑀 = σ න
𝑎
𝑏
(𝑓 𝑥 − 𝑔(𝑥))𝑑𝑥 = σ. 𝐴
The partition i has center of mass (xi*, yi*)
yi
*
Identify the position of xi* and yi*
xi* = xi and yi* = ½ (f(xi) - g(xi)) + g(xi)
= ½ (f(xi) + g(xi))
xi
*

12. sb. y
b
a
)
(x
f
y =
)
(x
g
y =
0 sb. x
xi
xi
)
(
)
( i
i x
g
x
f −
Ai
xi
*
yi
*
𝑀𝑦 = 𝜎 න
𝑎
𝑏
𝑥 𝑑𝐴 = 𝜎 න
𝑎
𝑏
𝑥(𝑓 𝑥 − 𝑔(𝑥))𝑑𝑥
Accumulate myi and mxi, take the limit,
then state in integral form:
𝑀𝑥 = 𝜎 න
𝑎
𝑏
𝑦 𝑑𝐴 = 𝜎 න
𝑎
𝑏
1
2
(𝑓 𝑥 + 𝑔(𝑥)). (𝑓 𝑥 − 𝑔 𝑥 )𝑑𝑥 =
𝜎
2
න
𝑎
𝑏
𝑓 𝑥 2
− 𝑔 𝑥 2
𝑑𝑥
So, the center of mass of
the plane will be:
=
𝜎 ‫׬‬𝑎
𝑏
𝑥. (𝑓 𝑥 − 𝑔 𝑥 )𝑑𝑥
𝜎𝐴
=
1
2
𝜎 ‫׬‬𝑎
𝑏
𝑓 𝑥 2
− 𝑔 𝑥 2
𝑑𝑥
𝜎𝐴

13. Find the center of mass of a region bounded by : f(x) = 4 – x2,
x = 0, and y = 0. (At first quadrant)
∆x 2 x
y
0
f(x) = 4 – x2
4
x
f(x)
M = s A
x* = x
My = M x* = s A x
M = s A
y* = ½ f(x)
Mx = M y* = s A ½ f(x)
M = s A =
16
3
𝜎
∆𝐴 = 𝑓 𝑥 ∆𝑥
𝐴 = න
0
2
𝑓 𝑥 𝑑𝑥

14. So, the center of mass is (3/4, 8/5)
+
+
∆𝑀𝑦 = 𝜎. 𝑥. 𝑓 𝑥 ∆𝑥
𝑀𝑦 = 𝜎 න
0
2
𝑥. 𝑓 𝑥 𝑑𝑥
= 𝜎 න
0
2
𝑥 4 − 𝑥2 𝑑𝑥
= 𝜎
= 𝜎 𝜎
∆𝑀𝑥 = 1
2
𝜎. 𝑓 𝑥 2∆𝑥
𝑀𝑥 = 1
2
𝜎 න
0
2
𝑓 𝑥 2 𝑑𝑥
= 𝜎
= 𝜎
= 𝜎
𝜎
=
4𝜎
ൗ
16
3 𝜎
=
3
4
=
ൗ
128
15 𝜎
ൗ
16
3 𝜎
=
8
5

15. Find the center of mass of a region bounded by parabola y2 = 10x and
line y = x, if Mass density σ = 1.
0
sb. y
sb. x
10
y2 = 10x
y = x
10
x* = x
y* = ½ [f(x) - g(x)] + g(x)
= ½ [f(x) + g(x)]
M = s A = 1. A = A
∆x
x
f(x)-g(x)

16. = 4
= 5
So, the center of mass is (4, 5)
∆𝑀𝑦 = 𝜎. 𝑥. ∆𝐴
= x.
𝑀𝑦 = න
0
10
𝑥 10. 𝑥
1
2 − 𝑥 𝑑𝑥
∆𝑀𝑥 = 𝜎. 𝑦. ∆𝐴
= 𝜎. 𝑥. (𝑓 𝑥 − 𝑔 𝑥 )∆𝑥 = 𝜎. 1
2
(𝑓 𝑥 + 𝑔 𝑥 )(𝑓 𝑥 − 𝑔 𝑥 )∆𝑥
MX

17. Center of Mass of
A Solid of
Revolution(1)
• A region bounded by y = f(x), x = a, x = b, and X-axis, is
rotated around X-axis.
• Approximate the volume of the partition, add up,
take a limit and state ini integral form:
V = r2h → Vi =  f(xi)2xi
V = lim   f(xi)2xi
• Partition Vi has a mass of mi :
mi = r Vi = r  f(xi)2xi
M = lim  r  f(xi)2xi
• Moment Mi to YOZ-plane (Myz) :
Myz = xi*rVi = xi*rf(xi)2xi
Myz = lim  xi* r  f(xi)2xi
• So, the center of mass is :
xi
xi
xi
y
z
x
)
(x
f
)
( i
x
f
r =
y
x
b
a = 0
b
a = 0
)
(x
f
y =


Note: here, ρ is mass density for 3-D object

18. Center of Mass of
A Solid of
Revolution (2)
• A region bounded by y = f(x), x = a = 0, x = b, and X-
axis, rotated around Y-axis.
• Using shell method:
V = 2rhr → Vi = 2xif(xi)xi
• Partition Vi has a mass of mi
mi = r Vi = 2rxif(xi)xi
• Moment Mi to the XOZ-plane (Mxz)
Mxz = yi* r Vi = yi* r 2xif(xi)xi = ½ f(xi) r 2xif(xi)xi
• So, the center of mass is :
a=0
x
b
x
x
)
(x
f
y =
f(x)
y
b’
r = x
x
h = f(x)
a=0
x
b
y
b’
z
Mxz
M = 2r

19. Center of Mass of
A Solid of
Revolution (3)
• A region bounded by y = f(x), x = a = 0, x = b, and X-
axis, rotated around Y-axis.
• Using washer method:
V = (R2-r2)h → Vi  (b2-f(yi)2)yi
• Partition Vi has a mass of mi
mi = r Vi = r(b2-f(yi)2)yi
• Moment Mi to the XOZ-plane (Mxz)
Mxz = yi* r Vi = yi* r(b2-f(yi)2)yi
• So:
a=0
x
b
2
x
y =
y
b’
y
r=x=f(y)
R = b
y
b’
a=0
z
x


Mxz

20. Find the center of mass of a region bounded by: y = 4 – x2, x =
0, and y = 0, rotated around X-axis, if Mass Density = 1.
∆x 2 x
y
0
Y = 4 – x2
4
x
y
M = r V
x* = x
Myz = M x* = r V x
M = r V = V

21. So, the center of mass is (5/8, 0)

22. Find the center of mass of a region bounded by: y = 4 – x2, x =
0, and y = 0, rotatetd around Y-axis, if mass density = 1.
∆x 2 x
y
0
Y = 4 – x2
4
x
y
M = r V
y* = ½ y
Mxz = M y* = r V ½ y
M = r V = V

23. So, the center of mass is (0, 4/3)
Mxz
Mxz
a
b

24. Exercise
2
y
x =
y
x −
= 6
2
Y
6
X
0
6
Calculate the center of mass of the solid of
revolution:
1. If the plane is rotated around the X-axis
2. If the plane is rotated around the Y-axis

25. 4
y
y = 2x
2
2
x
y =
x
Calculate the center of mass of the solid of
revolution:
1. If the plane is rotated around the X-axis
2. If the plane is rotated around the Y-axis

26. Moment of Inertia

27. • Moment of inertia (IL) of a mass with respect to a
line-L is the second moment of a plane to line-L. It
is the multiplication of the mass and the square of
the distance (perpendicular) between the elements
to L.
• Moment of inertia with respect to X-axis is Iy, and
Moment of inertia with respect to Y-axis is Ix
Moment of Inertia of A Plane
0 sb. x
sb. y
yi*
xi*
mi
𝐼𝑦 ≈ 𝑥1
∗2
. ∆𝑚1 + 𝑥2
∗2
. ∆𝑚2 + ⋯ + 𝑥𝑖
∗2
. ∆𝑚𝑖 + ⋯ + 𝑥𝑛
∗2
. ∆𝑚𝑛 ≈ ෍
𝑖=1
𝑛
𝑥𝑖
∗2
. ∆𝑚𝑖
𝐼𝑥 ≈ 𝑦1
∗2
. ∆𝑚1 + 𝑦2
∗2
. ∆𝑚2 + ⋯ + 𝑦𝑖
∗2
. ∆𝑚𝑖 + ⋯ + 𝑦𝑛
∗2
. ∆𝑚𝑛 ≈ ෍
𝑖=1
𝑛
𝑦𝑖
∗2
. ∆𝑚𝑖

28. Moment Inertia of A Plane
If n approaching infinity then :
M = σ A dM = σ dA
(σ = mass density for 2-D)
𝐼𝑥 = lim
𝑛→∞
෍
𝑖=1
𝑛
𝑦𝑖
∗2
∆𝑚𝑖 = න𝑦2𝑑𝑀
𝐼𝑦 = lim
𝑛→∞
෍
𝑖=1
𝑛
𝑥𝑖
∗2
∆𝑚𝑖 = න𝑥2𝑑𝑀
With the same steps as we did for center of mass equations,
try to generate the equations for moment inertia using integral.