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Question

A small sphere initially at rest, falls in to a viscous liquid. Due to drag, heat is produced. Then the relation between rate of production of heat and the radius of the sphere at terminal velocity, is best represented by:
  1. r5
  2. r6
  3. r3
  4. r2


Solution

The correct option is A r5

We have that
Terminal velocity, VT=2r2g9η(ρsρL) ...(i)
r radius of sphere 
g acceleration due to gravity 
y coefficient of viscosity
ρs density of sphere
ρL density of fluid.

& viscous Force (Drag force):
F=6πηrVT ...(ii) for [v=VT]
rate of production of heatdQdt=Rate of energy production
dQdt=Power(P)
Hence we can write as:
P=F×VT
P=6πηrVT×VT
Substituting from Eq (i) & (ii),
P=6πrη[2r2g9η(ρsρL)]2
Or, P=6πηr5[2g9η(ρsρL)]2
Pr5

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