Question

A string of a musical instrument was being tuned using a tuning fork of known frequency, $f_0=1024\text{Hz}$. The tuning fork and the string were set to vibrate together. Both vibrated together for $10$ second and no beat was heard . What prediction can be made regarding the frequency of the string?

• A. $f_{\text{string}}$ can be of any value.
• B. $1023.4\text{Hz}<f_{\text{string}}<1023.6\text{Hz}$
• C. $1023.9\text{Hz}<f_{\text{string}}<1024.1\text{Hz}$
• D. Given data is insufficient

Answer 1

The correct option is C $1023.9\text{Hz}<f_{\text{string}}<1024.1\text{Hz}$

Beat frequency $\Delta f$ must be less than 1 in $10\text{s}$.
$\Rightarrow \Delta f<0.1\text{Hz}$
Difference in frequencies of the two sources is less than $0.1\text{Hz}$.
$(1024-0.1)\text{Hz}<f_{\text{string}}<(1024+0.1)\text{Hz}$
$\therefore 1023.9\text{Hz}<f_{\text{string}}<1024.1\text{Hz}$