A bubble of radius R in water of density rho | vedclass

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(B) Let a bubble of radius $R$ expand such that its surface moves with speed $v$. Consider a spherical shell of radius $x$ $(x \ge R)$ and thickness $

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$2 \pi \rho R^{3} v^{2}$

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(B) Let a bubble of radius $R$ expand such that its surface moves with speed $v$. Consider a spherical shell of radius $x$ $(x \ge R)$ and thickness $dx$ in the water. Due to the incompressibility of water,the volume flow rate across any spherical surface must be constant. At the bubble surface (radius $R$),the flow rate is $4 \pi R^{2} v$. At a distance $x$ from the center,the flow rate is $4 \pi x^{2} v_{x}$,where $v_{x}$ is the velocity of the water at distance $x$. Equating these,$4 \pi x^{2} v_{x} = 4 \pi R^{2} v$,which gives $v_{x} = \frac{R^{2} v}{x^{2}}$. The mass of the spherical shell of thickness $dx$ is $dm = \rho (4 \pi x^{2} dx)$. The kinetic energy $dK$ of this shell is $dK = \frac{1}{2} dm v_{x}^{2} = \frac{1}{2} (4 \pi x^{2} \rho dx) \left( \frac{R^{2} v}{x^{2}} \right)^{2}$. Simplifying,$dK = 2 \pi \rho R^{4} v^{2} \frac{dx}{x^{2}}$. The total kinetic energy $K$ is the integral from $x = R$ to $x = \infty$: $K = \int_{R}^{\infty} 2 \pi \rho R^{4} v^{2} \frac{dx}{x^{2}} = 2 \pi \rho R^{4} v^{2} \left[ -\frac{1}{x} \right]_{R}^{\infty} = 2 \pi \rho R^{4} v^{2} \left( 0 - (-\frac{1}{R}) \right) = 2 \pi \rho R^{3} v^{2}$.

Column-$I$	Column-$II$
$(a)$ Rain drops move downwards with constant velocity.	$(i)$ Viscous liquids
$(b)$ Floating clouds at a height in air.	$(ii)$ Viscosity
	$(iii)$ Less density

$A$ bubble of radius $R$ in water of density $\rho$ is expanding uniformly at speed $v$. Given that water is incompressible,the kinetic energy of water being pushed is

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Column-$I$ Column-$II$

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