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Derive the expression for the terminal velocity of a sphere moving in a high viscous fluid using stokes force.

Answer» <html><body><p></p>Solution :Expression for terminal velocity : Consider a sphere of radius <a href="https://interviewquestions.tuteehub.com/tag/r-611811" style="font-weight:bold;" target="_blank" title="Click to know more about R">R</a> which falls freely through a highly viscous liquid of <a href="https://interviewquestions.tuteehub.com/tag/coefficient-920926" style="font-weight:bold;" target="_blank" title="Click to know more about COEFFICIENT">COEFFICIENT</a> of viscosity `eta`. Let the density of the material of the sphere be `<a href="https://interviewquestions.tuteehub.com/tag/rho-623364" style="font-weight:bold;" target="_blank" title="Click to know more about RHO">RHO</a>` and the density of the fluid be `sigma`. <br/> Gravitational force acting on the sphere. <br/> `F_(G) = mg = (4)/(3)pi r^(3)rho g` (downward force) <br/> Up thrust, `U = (4)/(3)pi r^(3) sigma g` (upward force) <br/> viscous force `<a href="https://interviewquestions.tuteehub.com/tag/f-455800" style="font-weight:bold;" target="_blank" title="Click to know more about F">F</a>= 6pi eta r v_(t)` <br/> At terminal velocity `v_(t)` <br/> downward force = upward force <br/> `F_(G) - U = F rArr (4)/(3) pi r^(3) rho g - (4)/(3) pi r^(3) sigma g = 6pi eta r v_(t)` <br/> `v_(t) = (2)/(9) xx (r^(2)(rho - sigma))/(eta) g rArr v_(t) <a href="https://interviewquestions.tuteehub.com/tag/prop-607409" style="font-weight:bold;" target="_blank" title="Click to know more about PROP">PROP</a> r^(2)` <br/> Here, it should be noted that the terminal speed of the sphere is directly proportional to the square of its radius. If `sigma` is greater than `rho`, then the term `(rho - sigma)` becomes negative leading to a negative terminal velocity. <br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/SUR_PHY_XI_V02_C07_E03_007_S01.png" width="80%"/></body></html>


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