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

Options

A. `r^4`
B. `r^3`
C. `r^5`
D. `r^2`

Answer

C

Explanation

Rate of heat produced `=` power loss against vicous force
`implies(dQ)/(dt)=F_vxxv_T`
where the terminal velocity `v_T=(2r^2)/(9eta)(rho-sigma)g`
and viscous force `F_v=6pietarv_T`
Hence `(dQ)/(dt)=7pietarv_Txxv_T` or `v_Talphar^2`
`implies(dQ)/(dt)proprv_T^2` but `v_Tpropr^2`
Hence `(dQ)/(dt)propr^5`