Question

A uniform wire of length $L$ and mass $M$ is stretched between two fixed points, keeping a tension $F. A $ sound of frequency $\mu$ is impressed on it. Then the maximum vibrational energy is existing in the wire when $\mu=$

• A. $\frac{1}{2} \sqrt{\frac{ ML }{ F }}$
• B. $\sqrt{\frac{ FL }{ M }}$
• C. $2 \times \sqrt{\frac{ FM }{ L }}$
• D. $\frac{1}{2} \sqrt{\frac{ F }{ ML }}$

Answer 1

Answer: (D)

$m \propto(F)^{a}(M)^{b}(L)^{c}$
$T ^{-1}=\left( MLT ^{-2}\right)^{ a }( M )^{ b }( L )^{ c }$
$T^{-1}=M^{a+b} L^{a+c} T^{-2 a}$
$a+b=0 \quad b=\frac{-1}{2}$
$a+c=0 \quad c=\frac{-1}{2}$
$-2 a=-1 \quad a=\frac{1}{2}$
$\mu=k F^{1 / 2} M^{-1 / 2} L^{-1 / 2}$
$\mu=k \sqrt{\frac{F}{M L}}$