Question

The wave speed on a string is 150 m/s when the tension is 85.0 N . What tension will give a speed of 177 m/s ?

Answer 1

The wave speed on a string is given by
  $v=\sqrt{\frac{T}{\mu}}$
where, T is the tension in the string and $\mu$ is linear mass density of the string. And so, we get the ratio in case of two wave speeds for the same string ( and so, the same $\mu$ ) is given by
  $\frac{v_2}{v_1}=\sqrt{\frac{T_2}{T_1}}$
$\Rightarrow T_2=\left (\frac{v_2}{v_1} \right )^2T_1$
And so, for the given values,
$v_1=150~m/s,~~v_2=177~m/s,~~T_1=85.0~N$
And so, we get the tension in the case when the speed is 177 m/s, is
$\Rightarrow T_2=\left (\frac{177}{150} \right )^2\times 85.0$
$\Rightarrow T_2=118.354~N$