Frequency $(n)$ of a tuning fork depends upon length $(l)$ of its prongs, density $(\mathrm{p})$ and Young's…

Frequency $(n)$ of a tuning fork depends upon length $(l)$ of its prongs, density $(\mathrm{p})$ and Young's modulus $(Y)$ of its material. Then frequency and Young' modulus will be related as
  1. $n \propto \sqrt{Y}$
  2. $n \propto Y$
  3. $n \propto \frac{1}{\sqrt{Y}}$
  4. $n \propto \frac{1}{Y}$

Solution

Let $n \propto l^{a} \rho^{b} Y^{c}$
Putting dimensions of all the quantities, we have $\left(T^{-1}\right) \propto L^{a}\left(M L^{-3}\right)^{b}\left(M L^{-1} T^{-2}\right)^{c}$
Equating powers of $M, L$ and $T$ on both sides, we get $b+c=0, a-3 b-c=0$ and $-2 c=-1$
which give $a=-1, b=-\frac{1}{2}$ and $c=\frac{1}{2}$. Thus $n \propto l^{-1} \rho^{1 / 2} Y^{1 / 2}$. /

Asked in: JEE Mains - Units and Dimensions - Test 3

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