Question

when a solid sphere moves through a liquid the liquid opposes is the motion with a force the magnitude of the force depends upon the coefficient of viscosity of the liquid the radius of the sphere find the expression of force using dimension formula

Answer 1

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HERE IS YOUR ANSWER,


> Consider the movement of a ball inside a viscous fluid:

1) Radius - r
2) Coefficient of Viscosity - η
3) Density of the ball - d
4) Density of the liquid - ρ
5) Acceleration due to Gravity - g

> The ball is subjected to the influence of three forces: they are the weight, upthrust and the viscous force - drag or liquid friction.

> Weight of the ball = mg = 4/3 πr³dg

> Upthrust on the ball by the liquid = v×ρ×g = 4/3×πr³×ρg.

> According to Stokes Law,
Viscous force = 6πηV, (where V is the velocity at a given a time).

> At the outset, the downward force, weight, is greater than the combination of the upward forces.

> So, initially, the ball accelerates. The viscous force, which depends of the velocity, however, keeps increasing.

> As a result, at some point, the net force on the ball becomes zero and the velocity of the ball becomes constant.

> It is the Terminal Velocity - Vt

> When the balls moves at the terminal velocity,

=> 4/3πr3d = 4/3πr3ρg + 6πηVt

=> Vt = 2(d - ρ)gr2 / 9η.



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Answer 2

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