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When a solid moves therough a liquid, th...

When a solid moves therough a liquid, the liquid opposes the miotioon with a force F. The magnitude of F depends on the coefficient of viscosity `eta` of the liquid, the radius r of the sphere aknd the speed v of the sphere. Assuming that F is proportional to different powers of these quantities, guess a formula for F using the method of dimension.

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Suppose the formula is `F=keta^a r^b v^c`.
Then, `MLT^-2=[ML^-1T^-1]^aL^b(L/T)^c`
`=M^aL^-(a+b+c) T^(-a-c)`
Equating the exponents of M,L and T from both sides, `
` a=1`
` -a+b+c=1`
` -a-c=-2`
Solving these `a=, b=1, and c=1.`
solving these `a=1, b=1, and c=1`.
Thus, the formula for `f is `F=Ketarv`.
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When an object moves through a fluid, as when a ball falls through air or a glass sphere falls through water te fluid exerts a viscous foce F on the object this force tends to slow the object for a small sphere of radius r moving is given by stoke's law, F_(w)=6pietarv . in this formula eta in the coefficient of viscosity of the fluid which is the proportionality constant that determines how much tangential force is required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular distance z from and immobile surface. the magnitude of the force is given by F=etaAv//z . For a viscous fluid to move from location 2 to location 1 along 2 must exceed that at location 1, poiseuilles's law given the volumes flow rate Q that results from such a pressure difference P_(2)-P_(1) . The flow rate of expressed by the formula Q=(piR^(4)(P_(2)-P_(1)))/(8etaL) poiseuille's law remains valid as long as the fluid flow is laminar. For a sfficiently high speed however the flow becomes turbulent flow is laminar as long as the reynolds number is less than approximately 2000. This number is given by the formula R_(e)=(2overline(v)rhoR)/(eta) In which overline(v) is the average speed rho is the density eta is the coefficient of viscosity of the fluid and R is the radius of the pipe. Take the density of water to be rho=1000kg//m^(3) Q. What is the viscous force on a glass sphere of radius r=1mm falling through water (eta=1xx10^(-3)Pa-s) when the sphere has speed of 3m/s?

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