Question

On a cello the 0.68 m long strings are turned by winding one end around a peg or fret. For a peg diameter of 15mm, calculate how many turns will be necessary to achieve a tension in the 1.36 mm diameter string of 84 N (approximately middle G). Assume the string is made of solid music wire (not a completely accurate assumption, but will work in this problem).

Answer 1

The freq of middle G is necessary (I found in one reference: 391.995 Hz), and the material of the strings (density).

If known, we have:

$\nu =\frac{1}{2l'}\sqrt{\frac{T}{\mu }}\: \left ( 1 \right )$

We have to determine l' from this eq, as the length for which that freq is emitted.

$l' =\frac{1}{2\nu }\sqrt{\frac{T}{\mu }}\: \left ( 2 \right )$

miu=mass of unit length

$\mu =\frac{m}{l}=\frac{\rho V}{l}=\rho A=\rho \frac{\pi d^{2}}{4}\: \left ( 3 \right )$

d=diameter of the string

Replace (3) in (2): $l'=\frac{1}{\nu d}\sqrt{\frac{T}{\pi \rho }}\: \left ( 4 \right )$

Numerically:........

The number of turns will be:

$N =\frac{l-l'}{\pi D}\: \left ( 5 \right )$

D=diameter of the peg