Question

A piano tuner stretches a steel piano wire with a tension of 765 n. the steel wire has a length of 0.700 m and a mass of 5.25 g . part a what is the frequency f1 of the string's fundamental mode of vibration?

Answer 1

We will use Mersenne's law that states:
$f=\frac{1}{2L}\sqrt{\frac{T}{\mu}$
Where f is fundamental frequency, T is the tension, $\mu$ is linear density(mass divided by length) and L is the length of the string. 
Let us find the linear density:
$\mu=\frac{m}{L}=\frac{5.25}{0.7\cdot1000}=0.0075\frac{kg}{m}$
Now we just have to plug in all the number in the formula:
$f=\frac{1}{2L}\sqrt{\frac{T}{\mu}}=\frac{1}{2\cdot 0.7}\sqrt{\frac{765}{0.0075}}=228.12 Hz$