A sonometer has a steel wire of diameter 0.5 mm…

(a) $r = 0.25\ \text{mm} = 2.5 \times 10^{-4}\ \text{m}$. $\mu = 7800\ \text{kg/m}^3 \times \pi \times (2.5 \times 10^{-4}\ \text{m})^2 = 1.53 \times 10^{-3}\ \text{kg/m}$, that is $1.53\ \text{g/m}$.

(b) $T = 4.0\ \text{kg} \times 9.8\ \text{m/s}^2 = 39.2\ \text{N}$. $v = \sqrt{\dfrac{39.2\ \text{N}}{1.53 \times 10^{-3}\ \text{kg/m}}} = 160\ \text{m/s}$. Then $f_1 = \dfrac{160\ \text{m/s}}{2 \times 0.60\ \text{m}} = 133\ \text{Hz}$.

(c) Twice the frequency needs four times the tension: $4 \times 4.0\ \text{kg} = 16\ \text{kg}$.