$A$ string fixed at both ends vibrates in three loops. The wavelength is $10 \, cm$. The length of the string is .... $cm$.

If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$, then the ratio of the final and initial fundamental frequencies is :

$A$ $1 \, cm$ long string vibrates with a fundamental frequency of $256 \, Hz$. If the length is reduced to $\frac{1}{4} \, cm$ keeping the tension unaltered,the new fundamental frequency will be: (in $, Hz$)

If the tension and diameter of a sonometer wire of fundamental frequency $n$ are doubled and density is halved,then its fundamental frequency will become

Two similar sonometer wires have fundamental frequencies of $500 \, Hz$. They are under the same tension. By what percentage should the tension be increased in one wire so that the two wires produce $5 \, \text{beats/sec}$ (in $\%$)?

Two identical strings of length $\ell$ and $2\ell$ vibrate with fundamental frequencies $N$ Hz and $1.5N$ Hz,respectively. The ratio of tensions for the smaller length to the larger length is