Question 6.1: The vertical displacement h of a freely falling body from it...
The vertical displacement h of a freely falling body from its point of projection at any time t is determined by the acceleration due to gravity g. Find the relationship of h with t and g by the use of Buckingham’s Pi theorem.
The above phenomenon can be described by the functional relation as
F(h, t, g)=0 (6.9)
Here the number of variables m = 3 (h, t, and g) and they can be expressed in terms of two fundamental dimensions L and T. Hence, the number of \pi terms = m – n = 3 – 2 = 1. In determining this \pi term, the number of repeating variables to be taken is 2. Since h is the dependent variable, the only choice left for the repeating variables is with t and g.
Therefore,
\pi_{1}=t^{a} g^{b} h (6.10)
By substituting the fundamental dimensions of the variables in the left and right hand sides of Eq. (6.10) we get
L ^{0} T ^{0}= T ^{a}\left( LT ^{-2}\right)^{b} L
Equating the exponents of T and L on both the sides of the above equation we have
a-2 b=0
and b+1=0
which give,
a=-2
b=-1
Hence, \pi_{1}=h / g t^{2}
Therefore the functional relationship (Eq. (6.9)) of the variables describing the phenomenon of free fall of a body under gravity can be written in terms of the dimensionless parameter \left(\pi_{1}\right.) as
f\left(\frac{h}{g t^{2}}\right)=0 (6.11)
From elementary classical mechanics we know that \frac{h}{g t^{2}}=\frac{1}{2} . One should know,
in this context, that the Pi theorem can only determine the pertinent dimensionless groups describing the problem but not the exact functional relationship between them.