A steel wire has a diameter of 0.50 mm. The density of steel is 7800 kg/m³…

Step 1. Mass of each metre. The radius is $0.25\ \text{mm} = 2.5 \times 10^{-4}\ \text{m}$. So $\mu = \rho\pi r^2 = 7800\ \text{kg/m}^3 \times \pi \times (2.5 \times 10^{-4}\ \text{m})^2 = 1.53 \times 10^{-3}\ \text{kg/m}$, or $1.53\ \text{g}$ in each metre.

Step 2. Speed. $v = \sqrt{\dfrac{40\ \text{N}}{1.53 \times 10^{-3}\ \text{kg/m}}} = 162\ \text{m/s}$.

Step 3. The new tension. $v^2 = T/\mu$, and we want $T$, so $T = \mu v^2 = 1.53 \times 10^{-3}\ \text{kg/m} \times (200\ \text{m/s})^2 = 61.3\ \text{N}$.