A mercury drop of radius 1 ~cm is sprayed int | vedclass

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(A) Given: Radius of big drop $R = 1 ~cm = 10^{-2} ~m$, number of small drops $n = 10^6$, surface tension $T = 460 	imes 10^{-3} ~N/m$.Conservation o

Vedclass · Accepted Answer

$0.057$

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(A) Given: Radius of big drop $R = 1 ~cm = 10^{-2} ~m$, number of small drops $n = 10^6$, surface tension $T = 460 	imes 10^{-3} ~N/m$.Conservation of volume: $n 	imes (\frac{4}{3} \pi r^3) = \frac{4}{3} \pi R^3 \implies r^3 = \frac{R^3}{n} \implies r = \frac{R}{n^{1/3}}$.For $n = 10^6$, $r = \frac{R}{(10^6)^{1/3}} = \frac{R}{10^2} = 10^{-2} R$.Energy expended $W = T 	imes \Delta A = T 	imes (n 	imes 4 \pi r^2 - 4 \pi R^2) = 4 \pi T (n r^2 - R^2)$.Substituting $r = R/100$: $W = 4 \pi T (n \frac{R^2}{10^4} - R^2) = 4 \pi R^2 T (\frac{n}{10^4} - 1)$.Since $n = 10^6$, $W = 4 \pi R^2 T (10^2 - 1) = 4 \pi R^2 T (99)$.Calculation: $W = 4 	imes 3.14 	imes (10^{-2})^2 	imes 460 	imes 10^{-3} 	imes 99$.$W = 4 	imes 3.14 	imes 10^{-4} 	imes 460 	imes 10^{-3} 	imes 99 \approx 0.057 ~J$.

$A$ mercury drop of radius $1 ~cm$ is sprayed into $10^6$ drops of equal size. The energy expended in joules is (Surface tension of mercury is $460 \times 10^{-3} ~N/m$)

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