Question

The frequency ($n$) of a tuning fork depends upon the length ($L$) of its prongs, the density ($d$) and Young's modulus ($Y$) of its material. It is given as $n\propto L^a{d}^b{Y}^c$. The values of $a,b$ and $c$ are :

• A. 1, 1/2, -1/2
• B. -1, - 1/2, 1/2
• C. 1/2, -1, -1/2
• D. 1/2, -1/2, 1

Answer 1

The correct option is B -1, - 1/2, 1/2

Given relation is $n\propto L^a{d}^b{Y}^c$
Dimensions of each quantity :
$L=L^1$
$d=M^1{L}^{-3}$
$Y=M^1{L}^{-1}{T}^{-2}$
$n=M^0{L}^0{T}^{-1}$
Putting the dimensions, we get $M^0{L}^0{T}^{-1}=M^{b+c}{L}^{a-3b-c}{T}^{-2c}$
Comparing both the sides :

We get $-2c=-1$ $⟹c=\frac{1}{2}$

Also, $b+c=0$ $⟹b=-c=\frac{-1}{2}$

From, $a-3b-c=0$
Or $a+\frac{3}{2}-\frac{1}{2}=$ $⟹a=-1$