Question Number 52859 by tanmay.chaudhury50@gmail.com last updated on 14/Jan/19
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jan/19
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jan/19
$${Here}\:{T}_{\mathrm{2}} >{T}_{\mathrm{1}} \:{so}\:{the}\:{string}\:{have}\:{a}\:{tendency}\:{to} \\ $$$${slide}\:{clockwise},{to}\:{balance}\:{friction}\:{acts}\:{to}\:{play} \\ $$$${As}\:{the}\:{difference}\:{of}\:{T}_{\mathrm{2}} \:{and}\:{T}_{\mathrm{1}} \:{increases} \\ $$$${friction}\:{also}\:{increases}\:{till}\:{it}\:{reach}\:{limiting} \\ $$$${friction},{then}\:{string}\:{starts}\:{slupping}. \\ $$$${considering}\:{small}\:{element}\:{of}\:{string}\:{in}\:{contact} \\ $$$${with}\:{pully}.{T}+{dT}\:{of}\:{right}\:{side}\:{is}\:{balanced} \\ $$$${by}\:{T}+{df}\:{of}\:{left}\:{side}.\:{df}={frictional}\:{force} \\ $$$${so}\:{T}+{dT}={T}+{df}\:\:\:\:\:{dT}={df} \\ $$$${from}\:{digram}… \\ $$$$\left({T}+{dT}\right){sin}\left(\frac{{d}\theta}{\mathrm{2}}\right)+\left({T}+{df}\right){sin}\left(\frac{{d}\theta}{\mathrm{2}}\right)={dN}\left({normal}\:{reaction}\right) \\ $$$$\mathrm{2}\left({T}+{dT}\right){sin}\left(\frac{{d}\theta}{\mathrm{2}}\right)={dN}\:\:\left[{since}\:{df}={dT}\right] \\ $$$$\mathrm{2}\left({T}+{dT}\right)\left(\frac{{d}\theta}{\mathrm{2}}\right)={dN} \\ $$$${Td}\theta+{dTd}\theta={dN}\:\:\left[{dTd}\theta={negligible}\right] \\ $$$${Td}\theta={dN} \\ $$$${df}=\mu{dN}\:\left[{friction}={coefficient}\:{of}\:{staticfriction}×{normal}\:{reaction}\right] \\ $$$${df}={dT}=\mu{dN}\:\:\:\:{so}\:\:{dN}=\frac{{dT}}{\mu} \\ $$$${Td}\theta=\frac{{dT}}{\mu} \\ $$$$\int_{{T}_{\mathrm{1}} } ^{{T}_{\mathrm{2}} } \frac{{dT}}{{T}}=\mu\int_{\mathrm{0}} ^{\theta} {d}\theta \\ $$$${ln}\left(\frac{{T}_{\mathrm{2}} }{{T}_{\mathrm{1}} }\right)=\mu\theta\:\: \\ $$$$\frac{{T}_{\mathrm{2}} }{{T}_{\mathrm{1}} }={e}^{\mu\theta} \:\:\:\:\:\:\:\:\:{T}_{\mathrm{2}} ={T}_{\mathrm{1}} {e}^{\mu\theta} \\ $$$${i}\:{have}\:{seen}\:{this}\:{in}\:{some}\:{book}\:{hence}\:{shared} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 14/Jan/19
$${A}\:{string}\:{is}\:{passing}\:{over}\:{the}\:{surface}\:{of}\:{a}\:{pully} \\ $$$${which}\:{is}\:{free}\:{to}\:{rotate}\:{about}\:{an}\:{axis}\:{passing} \\ $$$${through}\:{O},{the}\:{angle}\:{of}\:{contact}\:{of}\:{the}\:{string} \\ $$$${ln}\:{the}\:{pully}\:{is}\:\theta,{the}\:{static}\:{friction}\:{coefficient} \\ $$$${between}\:{pully}\:{and}\:{string}\:{is}\:\mu. \\ $$$${prove}\:{T}_{\mathrm{2}} ={T}_{\mathrm{1}} {e}^{\mu\theta} \:\:\:\:\left[{T}_{\mathrm{2}} >{T}_{\mathrm{1}} \right] \\ $$