Question

A particular guitar wire is 30.0 cm long and vibrates at a frequency of 196 Hz when no finger is placed on it. The next higher notes on the scale are 220 Hz, 247 Hz, 262 Hz and 294 Hz. How far from the end of the string must the finger be placed to play these notes?

• A. 26.7 cm, 23.8 cm, 22.4 cm, 20.0 cm
• B. 30 cm, 20 cm, 10 cm, 5 cm
• C. 23.8 cm, 22.4 cm, 20 cm, 10 cm
• D. none of these

Answer 1

The correct option is A

26.7 cm, 23.8 cm, 22.4 cm, 20.0 cm

Fundamental frequency and length are related as $f=\frac{v}{2L}$

Without holding the string the frequency is 196 Hz

$196=\frac{v}{2L}$

$\Rightarrow v=196\times 2L$

[On holding the string let's say effective length becomes L']

$f^{\prime }=\frac{v}{2L^{\prime }}$

$f^{\prime }=\frac{196\times 2L}{2\times L^{\prime }}$

$L^{\prime }=\frac{196\times L}{f^{\prime }}$

Given L = 30 cm

If $f^{\prime }=220Hz,L^{\prime }=26.7cm$

If $f^{\prime }=247Hz,L^{\prime }=23.8cm$

If $f^{\prime }=262Hz,L^{\prime }=22.4cm$

If $f^{\prime }=294Hz,L^{\prime }=20.0cm$