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what is the dependency of the rate of heat produced in terms of radius $r$ after the drop attains terminal velocity when A small ball of radius $r$ is falling in a viscous liquid under gravity?

My efforts include:

The forces acting on the sphere are its weight $\dfrac 4 3πr^3ρg$ downwards, buoyancy force $\dfrac 4 3 \pi r^3\sigma g$ upwards, and viscous force $6π\eta rv$ upwards. The sphere attains the terminal velocity it when the resultant force on it is zero i.e.,$$\dfrac 4 3πr^3ρg=\dfrac43πr^3σg+6πηrvt$$

I don't know how to Solve the above equation to get the terminal

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  • $\begingroup$ Hi Aakash Thoriya. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Jun 1 at 10:18
  • $\begingroup$ @Qmechanic okay $\endgroup$ – Aakash Thoriya Jun 1 at 10:28
  • $\begingroup$ @Qmechanic guide me, how can I solve this question? $\endgroup$ – Aakash Thoriya Jun 1 at 10:29
  • $\begingroup$ The heat generation is equal to the integral of the drag force integrated with respect to the differential displacement. $\endgroup$ – Chet Miller Jun 1 at 14:48

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