The bigger you are, the hotter you fall
A small sphere falls from rest in a viscous fluid. Due to viscous friction, heat is produced as the sphere moves relative to the fluid. The rate of heat production at terminal velocity is directly proportional to the power of the sphere's radius.
Use physical reasoning, scaling, dimensional analysis, or whatever tactics (apart from Google) you can think of to find .
The answer is 5.
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1 solution
Terminal Velocity V t = 9 η 2 . r 2 . g . ( ρ s − ρ L )
and viscous force F = 6 π η . r . V T
Rate of production of heat(power) [due to viscous force]
⇒ d t d Q = F . V T = ( 6 π η . r . V T ) ( V T ) = 6 π η . r . V T 2
⇒ 6 π η . r . 9 η 2 . r 2 . g . ( ρ s − ρ L ) 2 = 2 7 η 8 π . g 2 ( ρ s − ρ L ) 2 . r 5
⇒ d t d Q ∝ r 5