Question

In resonance tube experiment, the velocity of sound is given by \[v = 2{f_0}\left( {{l_2} - {l_1}} \right)\] . We found \[{l_1} = 25.0\;{\rm{cm}}\] and \[{l_2} = 75.0\;{\rm{cm}}\] . If there is no error in frequency, what will be the maximum permissible error in the speed of sound? (Take \[{f_0} = 325\;{\rm{Hz}}\])

Answer 1

Hint: The above problem can be resolved using the concept of instrumentation and error. The expression for the velocity of sound in the resonance tube is given. To find the minimum and the maximum value of sound, one must obtain the values by substituting the given values and applying conditions with the error calculations. Moreover, after receiving the desired maximum and the minimum values, the error can be calculated for the velocity.

Complete step by step answer:
The given expression for the speed of sound is,
\[v = 2{f_0}\left( {{l_2} - {l_1}} \right)..............\left( 1 \right)\]
Substituting the values in above equation to solve for minimum value as,
\[\begin{array}{l}
{v_{\min .}} = 2 \times 325\;{\rm{Hz}} \times \left( {75\;{\rm{cm}} - 25\;{\rm{cm}}} \right)\\
{v_{\min .}} = 32500\;{\rm{m/s}}
\end{array}\]
To obtain the maximum permissible error, the above expression for the speed of sound is differentiated as,
\[\begin{array}{l}
v = 2{f_0}\left( {{l_2} - {l_1}} \right)\\
dv = \dfrac{d}{{dt}}\left( {2{f_0}\left( {{l_2} - {l_1}} \right)} \right)\\
{v_{\max .}} = 2{f_0}\left( {d{l_2} - d{l_1}} \right)\\
{v_{\max .}} = 2{f_0}\left( { \pm \Delta {l_2} \pm \Delta {l_1}} \right)
\end{array}\]
Taking the positive value of above equation as,
\[{v_{\max .}} = 2{f_0}\left( {\Delta {l_2} + \Delta {l_1}} \right).......................\left( 2 \right)\]
Here, \[\Delta {l_1}\] and \[\Delta {l_2}\] are the values of last number of \[{l_1}\] and \[{l_2}\] respectively and their values are 0.1 cm.
Substituting the values in equation 2 as,
 \[\begin{array}{l}
{v_{\max .}} = 2 \times 325\;{\rm{Hz}} \times \left( {0.1\;{\rm{cm}} + 0.1\;{\rm{cm}}} \right)\\
{v_{\max .}} = 130\;{\rm{cm/s}}
\end{array}\]
 Therefore, the maximum permissible error in the speed of sound is \[V = \left( {32500 \pm `130} \right)\;{\rm{cm/s}}\].

Note:
In order to resolve the given problem, one must know the concepts and fundamentals to calculate the error given in any experimental value. The errors are the deviation of practical or the measured value from the standard value. The standard values are known in advance, while the calculated values are the ones that can be obtained using the devices and the mathematical calculations. Moreover, there are wide applications for the lab purpose to calculate the deviation from the standard value.