Question

Assuming that the period of vibration of a tuning fork depends upon the length $l$ of its prongs density $\rho$ of its material and the elastic constant $E$ of its material what would be the formula for the period of its vibration.

• A. $\frac{k}{l}\sqrt{\frac{y}{d}}$
• B. $\frac{2k}{l}\sqrt{\frac{y}{d}}$
• C. $\frac{k}{2l}\sqrt{\frac{2y}{d}}$
• D. $\frac{3k}{l}\sqrt{\frac{y}{2d}}$

Answer 1

The correct option is A $\frac{k}{l}\sqrt{\frac{y}{d}}$

Considering $n=k{l}^a{d}^b{y}^c$--------(A)

where k=dimensionless quantity

l=length $=L$; d=density$=kg/m^3$ and y=Young modulus$=kg/m{s}^2$

Substituting the above mentioned dimension in A we get,

$[{T]}^{-1}=[{L]}^a\left[{M{L}^{-3}]}^b\right[{M{L}^{-1}{T}^{-2}]}^c$

Comparing the power of $ML$ and $T$ we have,

$b+c=0$---------(1)

$a-3b-c=0$------(2)

$-2c=-1$

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

Putting the values of c in 1 we get,

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

Putting the values of b and c in 2 we get,

$a=-1$

Thus putting the values of a,b and c in A we get,

$n=k{l}^{-1}{d}^{1/2}{y}^{1/2}=\frac{k}{l}\sqrt{\frac{y}{d}}$