A spherical ball of radius ' $r$ ' and mass ' $m$ ' is dropped into a lake having liquid of density one-third of density of ball and viscosity $η$ Choose incorrect statement if at steady state its terminal speed is $V_{T}$

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Drag force at steady state is $\frac{2}{3} m g$

Gravitational potential energy looses at rate $m g v_{T}$ at steady state

Heat produced at rate $\frac{4}{3} m g v_{T}$ at steady state

Ball gains $\frac{63 v_{T}}{100}$ speed after a time $t = \frac{m}{6 π η r}$ from the instant of dropping

answer is C.

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Detailed Solution

At Steady State $F_{n e t} = 0 m g - F_{b} - F_{D} = 0$ , $m g - ρ_{l} V_{g} = F_{D}$ $m g - \frac{ρ_{S}}{3} V_{g} = F_{0}$
$F_{D} = \frac{2}{3} m g .$ Rate of Loss of gravitational P.E $\frac{d}{d x} m g h = m g v_{T}$
Rate of Heat Loss=Power of Drag Froce = $- F_{D} V = - \frac{2}{3} m g v$
$F_{n e t} = m g - F_{b} - 6 π η r v m a = m g - F_{b} - 6 π η r v, \frac{d v}{d t} = g - \frac{F_{b}}{m} - \frac{6 π η r}{m} v$
So time constant $τ = \frac{m}{6 π η r}$ At $t = τ, v' = (.63) v_{T}$