Question

A massless rod $BD$ of length $l$ is suspended by two identical massless strings $AB$ and $CD$ of equal lengths. A block of mass $m$ is suspended from a point $P$ such that $BP$ is equal to $x$ . If the fundamental frequency of the left wire is twice the fundamental frequency of the right wire, then the value of $\frac{l}{x}$ is

Answer 1

Answer: 5

$f\propto v\propto \sqrt{T}$
$f_{AB}=2f_{CD}$
$\therefore T_{AB}=4T_{CD}\dots$ (i)
Further $\Sigma {\tau }_p=0$
$\therefore (T{)}_{AB}(x)=(T{)}_{CD}(\ell -x)$
or $4x=l-x\left((T{)}_{AB}=4(T{)}_{CD}\right)$
or $x=l/5$
$\Rightarrow \frac{l}{x}=5$