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Step 1: Understand the Physical Context
A small sphere of radius $r$ falls through a viscous liquid. Due to the viscous force acting on the sphere, heat is produced. We are asked about the rate at which this heat is produced (power) when the sphere attains terminal velocity.
Step 2: Express the Viscous Force Using Stoke’s Law
According to Stoke’s law, the viscous force $F$ acting on a sphere moving at terminal velocity $V_T$ in a fluid of viscosity $\eta$ is:
$F = 6\pi \eta r V_T$
Step 3: Relate Power with Force and Velocity
Power (rate of production of heat) is given by the product of the force and velocity:
$\text{Power} = F \times V_T = 6 \pi \eta r V_T \times V_T = 6 \pi \eta r\, V_T^2$
Step 4: Formula for Terminal Velocity
For a sphere of radius $r$ falling in a viscous liquid, the terminal velocity $V_T$ is given by:
$V_T = \frac{2 r^2 (\rho - \sigma)\, g}{9\, \eta}$
where $\rho$ is the density of the sphere, $\sigma$ is the density of the fluid, $g$ is the acceleration due to gravity, and $\eta$ is the coefficient of viscosity of the fluid.
Clearly, $V_T \propto r^2$.
Step 5: Substitute $V_T$ into the Power Expression
Substituting $V_T \propto r^2$ into $\text{Power} = 6\pi \eta r\, V_T^2$, we see:
$\text{Power} \propto r \times (r^2)^2 = r \times r^4 = r^5$
Step 6: Conclude the Proportionality
Therefore, the rate of heat production (power dissipated due to viscous force) is proportional to $r^5$.
Correct Answer: $r^5$
