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TS EAMCET 7 May 2018 Shift 1 Solved Paper
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© examsnet.com
Question : 92
Total: 160
Consider a sphere of mass M and radius R centered at origin. The density of material of thesphere is ρ =
A
r
α
, where r is the radial distance, α and A are constants. If the inertia of the sphere about the axis passing through center is
6
7
M
R
2
, the value of α is
3
6
9
12
Validate
Solution:
Let an elemental spherical shell of radius r and thickness t .
The mass of the elemental spherical shell,
d
m
=
v
×
ρ
d
m
=
(
4
π
r
2
)
d
r
.
A
r
α
d
m
=
4
π
A
r
2
+
α
d
r
The mass of entire solid sphere,
M
=
4
π
A
R
∫
0
r
2
+
a
d
r
M
=
4
π
A
(
r
3
+
α
3
+
α
)
0
R
=
4
π
3
+
α
.
R
3
+
α
…… (I)
The moment of inertia of elemental spherical shell,
d
I
=
2
3
(
d
m
)
.
r
2
d
I
=
2
3
(
4
π
A
r
2
+
α
d
r
)
r
2
The moment of inertia of entire solid sphere,
I
=
R
∫
0
d
I
=
2
3
(
4
π
A
)
R
∫
0
r
4
+
α
.
d
r
I
=
2
3
4
π
A
(
r
5
+
a
5
+
α
)
0
R
I
=
2
3
4
π
A
(
R
5
+
a
5
+
α
)
I
=
2
3
(
4
π
A
3
+
α
R
3
+
α
)
R
2
(
3
+
α
)
5
+
α
From equation (I)
I
=
2
3
M
R
2
(
3
+
α
5
+
α
)
……(II)
The moment of inertia is given:
I
=
6
7
M
R
2
Substitute the value in equation ( 2 ).
6
7
M
R
2
=
2
3
M
R
2
(
3
+
α
5
+
α
)
3
+
α
5
+
α
=
9
7
α
=
−
12
© examsnet.com
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