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TS EAMCET 5 May 2018 Shift 2 Solved Paper
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© examsnet.com
Question : 94
Total: 160
The mass density inside a solid sphere of radius
r
varies as
ρ
(
r
)
=
ρ
0
(
r
R
)
β
,
where
ρ
0
and
β
are constants and
r
is the distance from the centre. Let
E
1
and
E
2
be gravitational fields due to sphere at distance
R
2
and
2
R
from the centre sphere. If
E
2
E
1
=
4
,
the value of
beta
is
2
2.5
3
4
Validate
Solution:
Consider the figure.
The mass enclosed in a sphere of radius r is calculated as,
M
r
=
∫
ρ
d
V
=
r
∫
0
ρ
0
(
r
R
)
β
.
4
π
r
2
d
r
=
[
4
π
ρ
0
R
β
.
r
β
+
3
β
+
3
]
0
r
Thus, the mass enclosed in a sphere of radius
R
2
is given by,
M
1
=
4
π
ρ
0
R
β
.
(
R
2
)
β
+
3
(
β
+
3
)
=
4
π
ρ
0
R
3
(
β
+
3
)
×
1
2
β
+
3
The mass enclosed in a sphere of radius R is given by,
M
2
=
4
π
ρ
0
R
β
.
R
β
+
3
(
β
+
3
)
=
4
π
ρ
0
R
3
(
β
+
3
)
Therefore, the gravitational field intensities are calculated as shown below.
E
1
=
G
M
1
(
R
2
)
2
=
4
G
R
2
×
4
π
ρ
0
R
3
(
β
+
3
)
×
1
2
β
+
3
And,
E
2
=
G
M
2
(
2
R
)
2
=
G
4
R
2
×
4
π
ρ
0
R
3
β
+
3
since,
E
2
E
1
=
4
.
Therefore,
G
4
R
2
×
4
π
ρ
0
R
3
β
+
3
4
G
R
2
×
4
π
ρ
0
R
3
β
+
3
×
1
2
β
+
3
=
4
2
β
+
3
16
=
4
2
β
+
3
=
64
Solve further to get the value of β as follows.
2
β
.
2
3
=
2
6
2
β
=
2
3
β
=
3
© examsnet.com
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