When the object falls through a viscous fluid with terminal velocity the total downward force becomes equal to total upward force
[tex]\\ \rm\rightarrowtail mg=U+F_v[/tex]
[tex]\\ \rm\rightarrowtail mg=m_1g+6\pi \eta rv_T[/tex]
[tex]\\ \rm\rightarrowtail (m-m_1)g=6\pi \eta rv_T[/tex]
[tex]\\ \rm\rightarrowtail v\rho g=6\pi \eta rv_T[/tex]
[tex]\\ \rm\rightarrowtail \dfrac{4}{3}\pi r^3(\rho-\sigma)g=6\pi\eta rv_T[/tex]
[tex]\\ \rm\rightarrowtail v_T=\dfrac{4gr^2(\rho-\sigma)}{18\eta}[/tex]
[tex]\\ \rm\rightarrowtail v_T=\dfrac{2}{9}\dfrac{gr^2(\rho-\sigma)}{\eta}[/tex]
