State stokes’ law. By using it deduce the expression for :
$(i)$ initial acceleration of smooth sphere and
$(ii)$ equation of terminal velocity of sphere falling freely through the viscous medium.
$(iii)$ Explain : Upward motion of bubbles produced in fluid.
State stokes’ law. By using it deduce the expression for :
$(i)$ initial acceleration of smooth sphere and
$(ii)$ equation of terminal velocity of sphere falling freely through the viscous medium.
$(iii)$ Explain : Upward motion of bubbles produced in fluid.
Scientist Stokes' said that, viscous force $\mathrm{F}_{(\mathrm{V})}$ on small spherical solid body of radius $r$ and moving with velocity $v$ through a viscous medium of large dimensions having coefficient of viscosity $\eta$ is $6 \pi \eta r v$.
As shown in figure a small spherical body of radius $r$, density $\rho$ falling in viscous medium of density $\sigma .$
Following forces acted on it.
$(i)$ Weight $\mathrm{F}_{1}=m g=$ (volume $\times$ density) $g$
$=\left(\frac{4}{3} \pi r^{3} \rho\right) g$
$\ldots$ $(1)$ (In downward)
Where $m=$ mass of sphere
$(ii)$ Buoyant force $\mathrm{F}_{2}=m_{\mathrm{o}} g$
$=\left(\frac{4}{3} \pi r^{3} \sigma\right) g$
$(2)$ (in upward)
where $m_{0}=$ mass of liquid having volume of sphere
$(iii)$ According to stokes' law
$\mathrm{F}_{(v)}=6 \pi \eta r v$
Resultant force acting on the sphere,
$\mathrm{F}_{\mathrm{R}}=\mathrm{F}_{1}-\mathrm{F}_{2}-\mathrm{F}_{(v)}$
$\mathrm{F}_{\mathrm{R}}=\left(\frac{4}{3} \pi r^{3} \rho\right) g-\left(\frac{4}{3} \pi r^{3} \sigma\right) g-6 \pi \eta r v$
$\mathrm{~F}_{\mathrm{R}}=\frac{4}{3} \pi r^{3} g(\rho-\sigma)-6 \pi \eta r v$
Similar Questions
A thin square plate of side $2\ m$ is moving at the interface of two very viscous liquids of viscosities ${\eta _1} = 1$ poise and ${\eta _2} = 4$ poise respectively as shown in the figure. Assume a linear velocity distribution in each fluid. The liquids are contained between two fixed plates. $h_1 + h_2 = 3\ m$ . A force $F$ is required to move the square plate with uniform velocity $10\ m/s$ horizontally then the value of minimum applied force will be ........ $N$
A thin square plate of side $2\ m$ is moving at the interface of two very viscous liquids of viscosities ${\eta _1} = 1$ poise and ${\eta _2} = 4$ poise respectively as shown in the figure. Assume a linear velocity distribution in each fluid. The liquids are contained between two fixed plates. $h_1 + h_2 = 3\ m$ . A force $F$ is required to move the square plate with uniform velocity $10\ m/s$ horizontally then the value of minimum applied force will be ........ $N$
A spherical ball of radius $1 \times 10^{-4} \mathrm{~m}$ and density $10^5$ $\mathrm{kg} / \mathrm{m}^3$ falls freely under gravity through a distance $h$ before entering a tank of water, If after entering in water the velocity of the ball does not change, then the value of $h$ is approximately:
(The coefficient of viscosity of water is $9.8 \times 10^{-6}$ $\left.\mathrm{N} \mathrm{s} / \mathrm{m}^2\right)$
A spherical ball of radius $1 \times 10^{-4} \mathrm{~m}$ and density $10^5$ $\mathrm{kg} / \mathrm{m}^3$ falls freely under gravity through a distance $h$ before entering a tank of water, If after entering in water the velocity of the ball does not change, then the value of $h$ is approximately:
(The coefficient of viscosity of water is $9.8 \times 10^{-6}$ $\left.\mathrm{N} \mathrm{s} / \mathrm{m}^2\right)$
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Give two uses of Stoke’s law.
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A small spherical solid ball is dropped in a viscous liquid. Its journey in the liquid is best described in the figure drawn by:-
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