MCQ
Two identical flutes produce fundamental notes of frequency $300 Hz$ at ${27^o} C.$ If the temperature of air in one flute is increased to ${31^o}$C, the number of the beats heard per second will be
- A$1$
- ✓$2$
- C$3$
- D$4$
✓
Answer
Correct option: B.
$2$
b
(b) Velocity of sound increases if the temperature increases.
(b) Velocity of sound increases if the temperature increases.
So with $v = n\lambda $, if $v$ increases $n$ will increase at
${27^o}C,\,{v_1} = n\lambda $, at ${31^o}C\,,\,\,{v_2} = (n + x)\lambda $
Now using $v \propto \sqrt T $ $(\because v= \sqrt{\frac{{\gamma \,R\,T}}{{M}}})$
$\frac{{{v_2}}}{{{v_1}}} = \sqrt {\frac{{{T_2}}}{{{T_1}}}} = \frac{{n + x}}{n}$
==> $\frac{{300 + x}}{{300}} = \sqrt {\frac{{(273 + 31)}}{{(273 + 27)}}} = \sqrt {\frac{{304}}{{300}}} = \sqrt {\frac{{300 + 4}}{{300}}} $
==> $1 + \frac{x}{{300}} = {\left( {1 + \frac{4}{{300}}} \right)^{1/2}} = \left( {1 + \frac{1}{2} \times \frac{4}{{300}}} \right)\,$ $[\because {(\,1 + x)^n} = 1 + nx]$
==> $x = 2.$
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