Question

The frequency $n$ of transverse waves in a string of length $l$ and mass per unit length $m$, under a tension $T$ is given by $n={kl}^a{T}^b{m}^c$ where $k$ is dimensionless. Then the values of $a,b,c$ are:

• A. $\frac{1}{2},\frac{1}{2},-\frac{1}{2}$
• B. $-1,\frac{1}{2},-\frac{1}{2}$
• C. $-\frac{1}{2},\frac{1}{2},\frac{1}{2}$
• D. $-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}$

Answer 1

Answer: (B)

$-1,\frac{1}{2},-\frac{1}{2}$

$l=M^0{L}^1{T}^0$

$m=M^1{L}^{-1}{T}^0$

$T=M^1{L}^1{T}^{-2}$

$n=T^{-1}$

$\Rightarrow M^0{L}^0{T}^{-1}=\left(L^1{)}^a\right(M^1{L}^1{T}^{-2}{)}^b(M^1{L}^{-1}{)}^c$

$\Rightarrow M^0{L}^0{T}^{-1}=M^{b+c}{L}^{a+b-c}{T}^{-2b}$

$\Rightarrow 2b=1,$ $\Rightarrow b=\frac{1}{2}$

$b+c=0,\Rightarrow c=-\frac{1}{2},a+b-c=0$

$a=-1$

Hence, option B is correct.