Question

The period of vibration of a tuning fork depends on the length l of its prong, density $\rho$ and Young's modulus 'Y' of its material. Then, the expression for the period of vibration will be.

• A. $k\rho \sqrt{\frac{l}{Y}}$
• B. $kl\sqrt{\frac{\rho }{Y}}$
• C. $\frac{k}{l}\sqrt{\frac{Y}{\rho }}$
• D. $\frac{l}{k}\sqrt{\frac{\rho }{Y}}$

Answer 1

Answer: (B)

$kl\sqrt{\frac{\rho }{Y}}$

As T depends on l, $\rho$ and Y
So, $T=k{l}^a{\rho }^b{Y}^c$ ...........(i)
where, $k=$ dimensionless constant
Substituting the dimension of each quantity in Eq.(i)
$\left[M^0{L}^0{T}^1\right]=k\left[M^0{L}^1{T}^0\right]a\left[M^1{L}^{-3}{T}^0\right]b[M^1{L}^{-1}{T}^{-2}{]}^c$
$\Rightarrow \left[M^0{L}^0{T}^1\right]=k\left[M^{b+c}{L}^{e-3b-c}{T}^{-2c}\right]$
Equating the powers of M, L and T on both sides
$b+c=0$
$\Rightarrow b=-c=\frac{1}{2}$
$a-2b-c=0$
$\Rightarrow a=2b=1$
$-2c=1$
$\Rightarrow c=-\frac{1}{2}$
Substituting the values of a, b and c in Eq. (i)
$T=k{l}^1{\rho }^{\frac{1}{2}}{Y}^{-\frac{1}{2}}$
$\therefore T=kl\sqrt{\frac{\rho }{Y}}$.