Question

The lowest note on a grand piano has a frequency of 27.5 Hz. The entire string is 2.00 m long and has a mass of 440g . The vibrating section of the string is 1.75m long.

What tension is needed to tune this string properly?

Answer 1

The frequency of vibrating string in its lowest mode is, \(v=\frac{v}{\lambda}=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}\)

Here, \(L\) is length of vibrating string, \(T\) is tension in the string, and \(\mu\) is linear mass density. \(\begin{aligned} \mu &=\frac{\text { mass of the string }}{\text { length of the string }} \\ &=\frac{0.44 \mathrm{~kg}}{2 \mathrm{~m}} \\ &=0.22 \mathrm{~kg} / \mathrm{m} \end{aligned}\)

Tension in the string, \(T=(2 L v)^{2} \mu\)

\(=4(1.75 \mathrm{~m})^{2}(27.5 \mathrm{~Hz})^{2}(0.22 \mathrm{~kg} / \mathrm{m})\)

\(=2038 \mathrm{~N}\)