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

Answer

D

Explanation

Terminal velocity `(v_T)=(2(r^2)g)/(9 eta)(rho_S-rho_L)`
and viscous force `F=6 pi eta rv_(T)`
Rate of production of heat (Power): as viscous force is the only dissipative force. Hence,
`(dQ)/(dt)=Fv_(T)=(6 pi eta rv_(T))(v_(T))`
`(6 pi etarv_(T)^2)`
`=6 pi etar{(2)/(9)(r^(2)g)/(eta) (rho_(S)-rho_(L))}^(2)`
`=(8 pi g^(2))/(27 eta) (rho_(S)-rho_(L))^(2)r^(5)` or `(dQ)/(dt)prop r^(5)`.