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Question-52859




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_2 >T_1  so the string have a tendency to  slide clockwise,to balance friction acts to play  As the difference of T_2  and T_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...  (T+dT)sin((dθ/2))+(T+df)sin((dθ/2))=dN(normal reaction)  2(T+dT)sin((dθ/2))=dN  [since df=dT]  2(T+dT)((dθ/2))=dN  Tdθ+dTdθ=dN  [dTdθ=negligible]  Tdθ=dN  df=μdN [friction=coefficient of staticfriction×normal reaction]  df=dT=μdN    so  dN=(dT/μ)  Tdθ=(dT/μ)  ∫_T_1  ^T_2  (dT/T)=μ∫_0 ^θ dθ  ln((T_2 /T_1 ))=μθ    (T_2 /T_1 )=e^(μθ)          T_2 =T_1 e^(μθ)   i have seen this in some book hence shared
$${Here}\:{T}_{\mathrm{2}} >{T}_{\mathrm{1}} \:{so}\:{the}\:{string}\:{have}\:{a}\:{tendency}\:{to} \\ $$$${slide}\:{clockwise},{to}\:{balance}\:{friction}\:{acts}\:{to}\:{play} \\ $$$${As}\:{the}\:{difference}\:{of}\:{T}_{\mathrm{2}} \:{and}\:{T}_{\mathrm{1}} \:{increases} \\ $$$${friction}\:{also}\:{increases}\:{till}\:{it}\:{reach}\:{limiting} \\ $$$${friction},{then}\:{string}\:{starts}\:{slupping}. \\ $$$${considering}\:{small}\:{element}\:{of}\:{string}\:{in}\:{contact} \\ $$$${with}\:{pully}.{T}+{dT}\:{of}\:{right}\:{side}\:{is}\:{balanced} \\ $$$${by}\:{T}+{df}\:{of}\:{left}\:{side}.\:{df}={frictional}\:{force} \\ $$$${so}\:{T}+{dT}={T}+{df}\:\:\:\:\:{dT}={df} \\ $$$${from}\:{digram}… \\ $$$$\left({T}+{dT}\right){sin}\left(\frac{{d}\theta}{\mathrm{2}}\right)+\left({T}+{df}\right){sin}\left(\frac{{d}\theta}{\mathrm{2}}\right)={dN}\left({normal}\:{reaction}\right) \\ $$$$\mathrm{2}\left({T}+{dT}\right){sin}\left(\frac{{d}\theta}{\mathrm{2}}\right)={dN}\:\:\left[{since}\:{df}={dT}\right] \\ $$$$\mathrm{2}\left({T}+{dT}\right)\left(\frac{{d}\theta}{\mathrm{2}}\right)={dN} \\ $$$${Td}\theta+{dTd}\theta={dN}\:\:\left[{dTd}\theta={negligible}\right] \\ $$$${Td}\theta={dN} \\ $$$${df}=\mu{dN}\:\left[{friction}={coefficient}\:{of}\:{staticfriction}×{normal}\:{reaction}\right] \\ $$$${df}={dT}=\mu{dN}\:\:\:\:{so}\:\:{dN}=\frac{{dT}}{\mu} \\ $$$${Td}\theta=\frac{{dT}}{\mu} \\ $$$$\int_{{T}_{\mathrm{1}} } ^{{T}_{\mathrm{2}} } \frac{{dT}}{{T}}=\mu\int_{\mathrm{0}} ^{\theta} {d}\theta \\ $$$${ln}\left(\frac{{T}_{\mathrm{2}} }{{T}_{\mathrm{1}} }\right)=\mu\theta\:\: \\ $$$$\frac{{T}_{\mathrm{2}} }{{T}_{\mathrm{1}} }={e}^{\mu\theta} \:\:\:\:\:\:\:\:\:{T}_{\mathrm{2}} ={T}_{\mathrm{1}} {e}^{\mu\theta} \\ $$$${i}\:{have}\:{seen}\:{this}\:{in}\:{some}\:{book}\:{hence}\:{shared} \\ $$
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 θ,the static friction coefficient  between pully and string is μ.  prove T_2 =T_1 e^(μθ)     [T_2 >T_1 ]
$${A}\:{string}\:{is}\:{passing}\:{over}\:{the}\:{surface}\:{of}\:{a}\:{pully} \\ $$$${which}\:{is}\:{free}\:{to}\:{rotate}\:{about}\:{an}\:{axis}\:{passing} \\ $$$${through}\:{O},{the}\:{angle}\:{of}\:{contact}\:{of}\:{the}\:{string} \\ $$$${ln}\:{the}\:{pully}\:{is}\:\theta,{the}\:{static}\:{friction}\:{coefficient} \\ $$$${between}\:{pully}\:{and}\:{string}\:{is}\:\mu. \\ $$$${prove}\:{T}_{\mathrm{2}} ={T}_{\mathrm{1}} {e}^{\mu\theta} \:\:\:\:\left[{T}_{\mathrm{2}} >{T}_{\mathrm{1}} \right] \\ $$

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