A gas bubble, from an exlosion under water, oscillates with a period T proportional to p^(a)d^(b)E^( c ). Where 'P' is the static pressure, 'd' is the density of water 'E' is the total energy of the explosion. Find the values of a, b and, c.
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Answer:
Given time period
T
T
∝
p
a
d
b
E
c
T
=
K
p
a
d
b
E
c
......(1)
where
K
is a constant of proportionality and dimensionless quantity.
Inserting the dimensions of Time, pressure, density and Energy in equation (1) we get
[
T
]
=
[
M
L
−
1
T
−
2
]
a
[
M
L
−
3
]
b
[
M
L
2
T
−
2
]
c
Equating powers of
M
,
L
,
and
T
on both sides we get
0
=
a
+
b
+
c
.....(2)
0
=
–
a
–
3
b
+
2
c
.....(3)
1
=
–
2
a
–
2
c
......(4)
Solving these equations
From (4)
a
+
c
=
−
1
2
.....(5)
Inserting this value in (2)
0
=
b
−
1
2
⇒
b
=
1
2
From (5)
a
=
−
1
2
−
c
Inserting values of
a
and
b
in (3)
0
=
−
(
−
1
2
−
c
)
−
3
×
1
2
+
2
c
⇒
3
c
=
1
⇒
c
=
1
3
Inserting value of
c
in (5)
a
+
1
3
=
−
1
2
⇒
a
=
−
1
2
−
1
3
⇒
a
=
−
5
6
∴
a
=
−
5
6
,
b
=
1
2
and
c
=
1
3
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