b
(b) For wire if

\(M =\) mass, \(\rho  =\) density, \(A = \)Area of cross section

\(V = \) volume, \(l =\) length, \(\Delta l\) = change in length

Then mass per unit length \(m = \frac{M}{l} = \frac{{Al\rho }}{l} = A\rho \)

And Young’s modules of elasticity

\(y = \frac{{T/A}}{{\Delta l/l}}\)

==> \(T = \frac{{Y\Delta lA}}{l}\).

Hence lowest frequency of vibration \(n = \frac{1}{{2l}}\sqrt {\frac{T}{m}} \)

\( = \frac{1}{{2l}}\sqrt {\frac{{y\left( {\frac{{\Delta l}}{l}} \right)A}}{{A\rho }}} = \frac{1}{{2l}}\sqrt {\frac{{y\Delta l}}{{l\rho }}} \)

==> \(n = \frac{1}{{2 \times 1}}\sqrt {\frac{{9 \times {{10}^{10}} \times 4.9 \times {{10}^{ - 4}}}}{{1 \times 9 \times {{10}^3}}}} = 35Hz\)