In mathematics, a planar lamina is a closed set in a plane of mass
m
and surface density
ρ
(
x
,
y
)
such that:
m
=
∫
∫
ρ
(
x
,
y
)
d
x
d
y
, over the closed set.
The center of mass of the lamina is at the point
(
M
y
m
,
M
x
m
)
where
M
y
moment of the entire lamina about the x-axis and
M
x
moment of the entire lamina about the y-axis.
M
y
=
lim
m
,
n
→
∞
∑
i
=
1
m
∑
j
=
1
n
x
i
j
∗
ρ
(
x
i
j
∗
,
y
i
j
∗
)
Δ
A
=
∬
x
ρ
(
x
,
y
)
d
x
d
y
, over the closed surface.
M
x
=
lim
m
,
n
→
∞
∑
i
=
1
m
∑
j
=
1
n
y
i
j
∗
ρ
(
x
i
j
∗
,
y
i
j
∗
)
Δ
A
=
∬
y
ρ
(
x
,
y
)
d
x
d
y
, over the closed surface.
Example 1.
Find the center of mass of a lamina with edges given by the lines
x
=
0
,
y
=
x
and
y
=
4
−
x
where the density is given as
ρ
(
x
,
y
)
=
2
x
+
3
y
+
2
.
m
=
∫
0
2
∫
x
4
−
x
(
2
x
+
3
y
+
2
)
d
y
d
x
Integrate 2x + 3y + 2 with respect to y and substitute the limits 4-x and x
m
=
∫
0
2
(
2
x
y
+
3
y
2
2
+
2
y
)
|
x
4
−
x
d
x
m
=
∫
0
2
(
[
2
x
(
4
−
x
)
+
3
(
4
−
x
)
2
2
+
2
(
4
−
x
)
]
−
[
2
x
(
x
)
+
3
(
x
)
2
2
+
2
(
x
)
]
)
d
x
m
=
∫
0
2
(
8
x
−
2
x
2
+
3
x
2
−
24
x
+
48
2
+
8
−
2
x
−
2
x
2
−
3
x
2
2
−
2
x
)
d
x
m
=
∫
0
2
(
8
x
−
2
x
2
+
3
2
x
2
−
12
x
+
24
+
8
−
2
x
−
2
x
2
−
3
2
x
2
−
2
x
)
d
x
m
=
∫
0
2
(
−
4
x
2
−
8
x
+
32
)
d
x
m
=
(
−
4
x
3
3
−
4
x
2
+
32
x
)
|
0
2
m
=
112
3
M
y
=
∫
0
2
∫
x
4
−
x
x
(
2
x
+
3
y
+
2
)
d
y
d
x
M
y
=
∫
0
2
(
2
x
2
y
+
3
x
y
2
2
+
2
x
y
)
|
x
4
−
x
d
x
M
y
=
∫
0
2
(
−
4
x
3
−
8
x
2
+
32
x
)
d
x
M
y
=
(
−
x
4
−
8
x
3
3
+
16
x
2
)
|
0
2
M
y
=
80
3
M
x
=
∫
0
2
∫
x
4
−
x
y
(
2
x
+
3
y
+
2
)
d
y
d
x
M
x
=
∫
0
2
(
x
y
2
+
y
3
+
y
2
)
|
x
4
−
x
d
x
M
x
=
∫
0
2
(
−
2
x
3
+
4
x
2
−
40
x
+
80
)
d
x
M
x
=
(
−
x
4
2
+
4
x
3
3
−
20
x
2
+
80
x
)
|
0
2
M
x
=
248
3
center of mass is at the point
(
80
3
112
3
,
248
3
112
3
)
=
(
5
7
,
31
14
)
Planar laminas can be used to determine moments of inertia, or center of mass.
