Question

A metal sphere of density and a radius is is falling through

Answer 1

The forces acting on the sphere are :

Weight = 4/3πr³рg downwards

Buoyant force = 4/3πr³бg upwards

Viscous force = 6πηrν upwards

The sphere attains the terminal velocity v₍t₎ when the resultant force acting on it is zero. i-e,

4/3πr³рg = 4/3πr³бg + 6πηrv₍t₎

Solving above equation gives us the terminal velocity,

v₍t₎ = 2r²(р-б)g / 9η

The rate of heat generation is equal to the rate of work done by the viscous force which in turn is equal to its power. Thus,

dQ/dt = (6πηrv₍t₎) v₍t₎ = 8π(р-б)²g²r⁵ / 27η