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

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

$(b)$ Let bubble of radius $x$ expands and pushes water of density $\rho$ by a distance $d x$. Then, kinetic energy of water of strip $d x$ is

$d K

Vedclass · Accepted Answer

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

Vedclass · Answer

$(b)$ Let bubble of radius $x$ expands and pushes water of density $\rho$ by a distance $d x$. Then, kinetic energy of water of strip $d x$ is

$d K=\frac{1}{2} d m \cdot v_{x}^{2} \Rightarrow d K=\frac{1}{2} 4 \pi x^{2} \cdot d x \cdot \rho \cdot v_{x}^{2}$

or $d K=2 \pi R^{2} \cdot \rho \cdot v_{x}^{2} \cdot d x \quad \ldots$ $(i)$

Considering laminar flow, for section $1$ and $2$ shown in figure, by equation of continuity

$A_{1} v_{1}=A_{2} v_{2} \Rightarrow 4 \pi x^{2} v_{x}=4 \pi R^{2} v$

$\Rightarrow \quad v_{x} =\frac{R^{2}}{x^{2}} \cdot v\quad \ldots$ $(ii)$

From Eqs. $(i)$ and $(ii)$, we have

$d K=2 \pi x^{2} \rho\left(\frac{R^{2}}{x^{2}} \cdot v \cdot\right)^{2} d x$

$\Rightarrow d K=2 \pi R^{4} \rho v^{2} \cdot\left(\frac{d x}{x^{2}}\right)$

So, kinetic energy of complete volume of water,

$K=\int \limits_{R}^{\infty} d K=2 \pi R^{4} \rho v^{2} \int \limits_{R}^{\infty} \frac{d x}{x^{2}}$

$\Rightarrow \quad K=2 \pi R^{4} \cdot \rho \cdot v\left[-\frac{1}{x}\right]_{R}^{\infty}$

$=2 \pi \rho R^{3} v^{2}$

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

Similar Questions

Mumbai needs $1.4 \times 10^{12} L$ of water annually. Its effective surface area is $600 \,km ^2$ and it receives an average rainfall of $2.4 \,m$ annually. If $10 \%$ of this rain water is conserved, it will meet approximately

A wooden block, with a coin placed on its top, floats in water as shown in fig. the distance $l $ and $h$ are shown there. After some time the coin falls into the water. Then

A vessel contains oil (density =$ 0.8 \;gm/cm^3$) over mercury (density = $13.6\; gm/cm^3$). A homogeneous sphere floats with half of its volume immersed in mercury and the other half in oil. The density of the material of the sphere in $ gm/cm^3$ is

A hollow spherical shell at outer radius $R$ floats just submerged under the water surface. The inner radius of the shell is $r$. If the specific gravity of the shell material is $\frac{27}{8}$ $w.r.t.$ water, the value of $r$ is$......R$