Question

The period of vibration of a tuning fork depends on length $\left(l\right)$ of its prongs, density $d$ and Young's modulus $\left(Y\right)$ of its material. Determine an expression for period of vibration using the method of dimensions.

Answer 1

Freq f = $[T{]}^{-1}$

length of the arm L = [ L]

density $\rho$ = $\left[M\right][L{]}^{-3}$

young' s modulus = Y =$\left[M\right]\left[L{]}^{-1}\right[T{]}^{-2}$

let $f=L^a{\rho }^b{Y}^c$

so a - 3b - c = 0

b + c = 0

C = $\frac{1}{2}$

b = $\frac{-1}{2}$

a = -1

f = $\frac{1}{L}\times \sqrt{\frac{y}{\rho }}$