Question Number 226538 by mr W last updated on 02/Dec/25
Answered by mr W last updated on 07/Dec/25
$${length}\:{of}\:{cylinder}\:={L} \\ $$$${cross}−{section}\:=\mathrm{2}{RL} \\ $$$${at}\:{speed}\:{v} \\ $$$${Mv}=\left({M}+\mathrm{2}{RLnmvdt}\right)\left({v}+{dv}\right) \\ $$$${Mdv}+\mathrm{2}{RLnmv}^{\mathrm{2}} {dt}=\mathrm{0} \\ $$$$\frac{{dv}}{{v}^{\mathrm{2}} }=−\frac{\mathrm{2}{RLnm}}{{M}}{dt} \\ $$$$\frac{\mathrm{1}}{{v}_{\mathrm{0}} }−\frac{\mathrm{1}}{{v}}=−\frac{\mathrm{2}{RLnmt}}{{M}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{{v}}=\frac{\mathrm{1}}{{v}_{\mathrm{0}} }+\frac{\mathrm{2}{RLnmt}}{{M}} \\ $$$${say}\:{at}\:{t}={t}_{\mathrm{1}} :\:{v}=\frac{{v}_{\mathrm{0}} }{\mathrm{2}} \\ $$$$\frac{\mathrm{2}}{{v}_{\mathrm{0}} }=\frac{\mathrm{1}}{{v}_{\mathrm{0}} }+\frac{\mathrm{2}{RLnmt}_{\mathrm{1}} }{{M}} \\ $$$$\Rightarrow{t}_{\mathrm{1}} =\frac{{M}}{\mathrm{2}{RLnmv}_{\mathrm{0}} } \\ $$