Find the moment of inertia of a right circular cone of base radius 1 and height 1 about an axis through the vertex parallel to the base.
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The mass of the elemental disc is
d
m
=
ρ
⋅
π
r
2
d
z
The density of the cone is
ρ
=
M
V
=
M
1
3
π
R
2
h
Therefore,
d
m
=
M
1
3
π
R
2
h
π
r
2
d
z
d
m
=
3
M
R
2
h
r
2
d
z
But
R
r
=
h
z
r
=
R
z
h
d
m
=
3
M
R
2
h
⋅
R
2
h
2
⋅
z
2
d
z
=
3
M
h
3
z
2
d
z
The moment of inertia of the elemental disc about the
z
−
axis is
d
I
=
1
2
d
m
r
2
d
I
=
1
2
⋅
3
M
h
3
z
2
⋅
z
2
R
2
h
2
d
z
d
I
=
3
2
⋅
M
R
2
h
5
z
4
d
z
Integrating both sides,
I
=
3
2
⋅
M
R
2
h
5
∫
h
0
z
4
d
z
I
=
3
2
⋅
M
R
2
h
5
[
z
5
5
]
h
0
I
=
3
2
⋅
M
R
2
h
5
⋅
h
5
5
=
3
10
M
R
2
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