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An adiabatic piston of mass m equally divides an insulated container of total volume v_(0) and length l. A light spring connect the piston to right wall. The container has helium gas. The pressure on both side of piston is P_(0). The container starts moving with acceleration 'a' towards right, find the stretch of the spring when acceleration of the piston equals acceleration of container. (Assume displacement of the piston lt lt l) |
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Answer» SOLUTION :Free BODY diagram of the separator `P_(2)A+ma=P_(1)A+kx` ….(`1`) where `A` is the area of cross SECTION of separator and `x` is the stretch of the spring We can write `P_(0)((V_(0))/(2))^(gamma)=P_(1)((V_(0))/(2)-Ax)^(gamma)` …..(`2`) and `P_(0)((V_(0))/(2))^(gamma)=P_(2)((V_(0))/(2)+Ax)^(gamma)` ....(`3`) from equation `P_(1)=P_(0)((V_(0))/((V_(0)-2Ax)))^(gamma)=P_(0)((1)/(1-(2Ax//V_(0))))^(gamma)` `P_(1)=P_(0)(1-(2Ax)/(A(l//2)))^(-gamma)=P_(0)(1+(4xgamma)/(l))`.......(`4`) `(XLT lt l)` Similarly `P_(2)=P_(0)(1-(4xgamma)/(l))`.....(`5`) Substituting the values of `P_(1)` and `P_(2)` in equation (`1`) `P_(0)(1-(4xgamma)/(l))A+ma=P_(0)(1+(4xgamma)/(l))A+kx` `[k+(8gammaP_(0)A)/(l)]x=ma` `rArr x=[(ma)/(k+(8gammaP_(0)V_(0))/(l^(2)))]` 
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