Question

A small sphere falls from rest in a viscous liquid. Due to frication, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity.

Answer 1

Correct Answer - `(dQ)/(dt)propr^(5)`
Terminal velocity `v_(T) = (2r^(2)g)/(9eta)(rho_(s) - rho_(L))`
and viscous force `F = 6pirv_(T)`
Viscous force is the dissipative force. Hence.
`(dQ)/(dt) = Fv_(T) = (6pietarv_(T))(v_(T)) = 6pietarv_(T)^(2)`
`= 6pietar{(2)/(9)(r^(2)g)/(eta)(rho_(s) - rho_(L))}^(2) = (8pig^(2))/(27eta)(rho_(s) - rho_(L))^(2)r^(5) = (dQ)/(dt)propr^(5)`