104
6.5
(
A
)
( , , , , )
.
The units on the variables on the rhs are as follows:
V
fdlg
wm
=
1
2
[
]
,
[
]
[
]
[
]
[]
L
ML
d LlL
gT
T
T

==
===
Because mass
M
occurs in only one term, it cannot enter the relationship.
6.6
[ ]
[ ] [ ]
[ ]
V
f
V
L
T
L
M
L
M
LT
=
=
=
=
=
( ,
,
).
,
,
,
.
l
l
r m
r
m
3
∴
There is one
p

term:
p
r
m
1
=
V
l
.
∴
=
=
∴
=
=
p
p
r
m
1
1
2
0
f
V
C
(
)
,
Re
Const.
or
Const.
l
6.7
[ ]
[ ]
[ ]
[ ]
V
f
d
V
L
T
M
T
M
L
d
L
=
=
=
=
=
(
,
,
).
,
,
,
.
s r
s
r
2
3
∴
=
∴
=
=
∴
=
p
s
r
p
p
s
r
1
2
1
1
2
0
2
V d
f
C
V d
C
.
(
)
,
onst.
or We = Const.
6.8
[ ]
[ ]
[ ] [ ]
V
f H g m
V
L
T
g
L
T
m
M
H
L
=
=
=
=
=
(
,
,
,
,
,
.
2
∴
=
∴
=
∴
=
p
p
1
0
2
1
gHm
V
C
V
gH
C
.
.
/
.
6.9
[ ]
[ ]
[ ]
[ ] [ ]
[ ]
V
f H g m
V
L
T
H
L
g
L
T
m
M
M
L
M
LT
=
=
=
=
=
=
=
(
,
,
,
, ).
,
,
,
,
,
.
r m
r
m
2
3
Choose repeating variables
H g
,
,
r
(select ones with simple dimensionswe couldn’t
select
V, H,
and
g
since
M
is not contained in any of those terms):
p
r
p
r
p
m
r
1
2
3
1
1
1
2
2
2
3
3
3
=
=
=
VH
g
mH
g
H
g
a
b
c
a
b
c
a
b
c
,
,
.
∴
=
=
=
=
=
p
r
p
r
p
m
r
m
r
1
0
2
3
3
3 2
3
V
g
H
V
gH
m
H
gH
gH
.
.
.
/
∴
=
V
gH
f
m
H
gH
1
3
3
r
m
r
,
Note: The above dimensionless groups are formed by observation: simply
combine the
dimensions so that the
p

term is dimensionless. We could have set up equations similar
to those of Eq. 6.2.11 and solved for
11
1
22
2
333
, ,
and , ,
c
and , , .
ab
c
a
b
abc
But the
method of observation is usually successful.
6.10
[ ]
[ ] [ ]
[ ]
[ ]
F
f d
V
F
ML
T
d
L
V
L
T
M
LT
M
L
D
D
=
=
=
=
=
=
( , ,
,
,
).
,
,
,
,
.
l
m r
m
r
2
3