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The pressure wave $P = 0.01 \sin [1000 t - 3 x] {Nm}^{- 2}$ correspond to the sound produced by a vibrating blade on a day when atmospheric temperature is $0^{0} C$ . On some other day when temperature is T, the speed of sound produced by the same blade and at the same frequency is found to be $336 {ms}^{- 1}$ . Approximate value of T in ⁰C is...........

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Detailed Solution

Given, $P = 0.01 \sin (1000 t - 3 x) N / m^{2}$

$[∴ P = P_{0} \sin (ω t - k x)]$

Here, $ω = 1000, k = 3$

$V = \frac{ω}{K} \Rightarrow V = \frac{1000}{3}$

We , $V = \sqrt{\frac{r R T}{M}} \Rightarrow V \propto \sqrt{T}$

$\Rightarrow \frac{V_{2}}{V_{1}} = \sqrt{\frac{T_{2}}{T_{1}}}$

$\Rightarrow \frac{{V_{2}}^{2}}{{V_{1}}^{2}} = \frac{T_{2}}{T_{1}} \Rightarrow T_{2} = \frac{{(V_{2})}^{2}}{{(V_{1})}^{2}} \times T_{1}$

$\Rightarrow T_{2} = \frac{{(336)}^{2}}{{(\frac{1000}{3})}^{2}} \times 273 = 277.38 K$

$\Rightarrow T_{2} = {4.38}^{0} C = 4^{0} C$

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