A small sphere of radius $r$ falls from rest in a viscous liquid. As a result, heat is produced due to the vis

Question

A small sphere of radius $r$ falls from rest in a viscous liquid. As a result, heat is produced due to the viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional to:

Accepted Answer

1. **Terminal Velocity ($v_t$):** When a sphere of radius $r$ falls through a viscous fluid, it attains a constant terminal velocity given by the formula $v_t = \frac{2r^2(ho - \sigma)g}{9\eta}$ . This implies that $v_t \propto r^2$.
2. **Viscous Force ($F$):** According to Stokes' Law, the viscous drag force acting on the sphere moving with terminal velocity is $F = 6\pi \eta r v_t$ . Substituting the proportionality for $v_t$, we get $F \propto r(r^2) \propto r^3$.
3. **Rate of Heat Production (Power):** The rate of heat production is equal to the power ($P$) dissipated by the viscous force, which is the product of the force and velocity: $P = F 	imes v_t$.
4. **Calculation:** Substituting the proportionalities:
   $$ P \propto (r^3) 	imes (r^2) $$
   $$ P \propto r^5 $$
   Thus, the rate of heat production is proportional to the fifth power of the radius.

Step-by-Step Solution

Practice All PHYSICS Questions