Page 38 - Aeronautical Engineer Data Book

Fundamental dimensions and units 25
2
–3 c
a
–1 d
–1 b
T –1 = L (LT ) (ML ) (L T )
a
–b
b
c
2d
= L L T M L –3c L T –d
In order for the equation to balance:
For M, c must = 0
For L, a + b –3c + 2d = 0
For T, –b –d = –1
Solving for a, b, c in terms of d gives:
a = –1 –d
b = 1 –d
Giving
0
n = d (–1 –d) V (1 –d) # d
Rearranging gives:
nd/V = (Vd/ )X
Note how dimensional analysis can give the
‘form’ of the formula but not the numerical
value of the undetermined constant X which, in
this case, is a compound constant containing the
original constant Y and the unknown index d.

2.8 Essential mathematics

2.8.1 Basic algebra
m
n
a 2 a = a
m+n
m
n
a 4 a = a m–n
m n
(a ) = a mn
n m m/n
� = a
�a
1
3 = a –n
a n
o
a = 1
n m p
np
(a b ) = a b mp
�� n = 3
n
a
a
3
n
b
b
n � n � n �
�ab = �a 2 �b
�
n �a
n 3 3
� =
�a\b
n �
�b