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Byju's Answer
Standard XII
Physics
Variation of G Due to Height and Depth
A straight tu...
Question
A straight tunnel is dug into the earth as shown in figure at a distance
b
from its center. A ball of mass
m
is dropped from one of its ends. Find the time it takes to reach the other end is approximately.
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Solution
Potential energy at start of tunnel (1)
u
1
=
G
M
m
R
e
potential energy (2)
u
2
=
G
⎛
⎜ ⎜ ⎜
⎝
M
4
3
π
R
3
e
⎞
⎟ ⎟ ⎟
⎠
(
4
3
π
b
3
)
×
m
G
M
m
b
2
R
3
e
Energy lost
G
M
m
R
e
−
G
M
m
b
2
R
3
e
G
M
m
R
e
[
1
−
b
2
R
2
e
]
[But this is an SHM motion as
d
2
E
d
x
2
=
0
]
G
M
m
R
e
[
1
−
b
2
R
2
e
]
=
1
2
k
x
2
x
=
√
R
2
e
−
b
2
G
M
m
R
e
[
1
−
b
2
R
2
e
]
=
1
2
k
(
R
2
e
−
b
2
)
k
=
2
G
M
m
R
3
e
2
π
T
=
√
k
m
⇒
T
=
2
π
√
m
k
T
=
π
⎷
m
2
G
M
m
R
3
e
=
2
π
√
R
3
e
2
G
M
=
2
π
√
R
e
2
g
Time taken to reach other end is
2
π
√
R
e
2
g
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