The period of vibration of a tunign fork depends on the length I of its prong, density d and Young's modulus Y of the meterial. Deduce an expression for the period of vibration (T) using dimensional analysis.

Explanation

Let `T = K I^a d^b Y^c ….(i)`
where K is dimensionless constant.
`[M_0L^0T^1] = L^a [ML^(-3)]^b [ML^(-1)T^(-2)]^c`
`=M^(b +c) L^(a -3b - c) T^(-2c)`
Applying the principle of homogeneity of
dimensions, we get
b +c = 0 …(ii)
a -3b - c = 0 .....(iii)
`=-2c = 1, c = - (1)/(2)`
From (ii), `b = -c = (1)/(2)`
From (iii), `a = 3b +c = (3)/(2) - (1)/(2) =1`
Putting these values in (i), we get
`T = K I^1 d^(1//2) Y^(-1//2) = KI sqrt(d//Y)`