A small sphere of radius 2 cm falls from rest in a viscous liquid. Heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional to

Options

A. `2^(2)`
B. `2^(3)`
C. `2^(4)`
D. `2^(5)`

Answer

D

Explanation

Terminal velocity `v_(t)=(2r^(2)(rho-sigma)g)/(9eta)`
`v_(t)propr^(2)`
The rate of production of heat is
`(DeltaH)/(Deltat)=Fv_(t)^(2)`
`F=6pietar`
`therefore(DeltaH)/(Deltat)=6pietar[(2(rho-sigma)r^(2)g)/(9eta)]^(2)`
`=6pietar[(4(rho-sigma)^(2)r^(4)g^(2))/(81eta^(2))]`
`=(24pieta(rho-sigma)^(2)g^(2)r^(5))/(81eta^(2))`
`=[(8pi(rho-sigma)^(2)g^(2))/(27eta)]r^(5)`
`therefore(DeltaH)/(Deltat)propr^(5)`
But r = 2 cm
`therefore(DeltaH)/(Deltat)prop2^(5)`