The frequency of vibration of a string depends on the length L
between the nodes, the tension F in the string and its mass per unit
length m. Guess the expression for its frequency from dimensional
analysis.
The frequency depends on the length L, mass per unit length m and tension F
f LaFbmc
Let us assume a dimensionless constant k such that
f = k LaFbmc
Dimensions of f = [T-1]
Dimensions of L = [L]
Dimensions of F = [MLT-2]
Dimensions of m = [ML-1]
Dimensionally
[T-1] = [L]a [MLT-2] b [ML-1]c
= [Mb+c La+b-c T-2b]
On comparing we can see that
b + c =0 …(i)
a + b – c =0 …(ii)
-2b = -1 …(iii)
On solving (i), (ii) and (iii)
a = -1, b = 1/2 and c = -1/2
So frequency can be expressed as
f = k L-1 F1/2 m-1/2
f =