A resistance coil of resistance r connected to an external battery, is placed inside an adiabatic cylinder fitted with a frictionless platon of mass m and same area A. Initially cylinder contains one mole of ideal gas he. A current through the coil such that temperature of gas varies as `T=T_(0)+"at"+"bt"^(2),` keeping pressure constant with time t. Atosphere pressure above piston is `P_(0)`. Find

(a) Current l flowing through the coil as function of time and
(b) Speed of piston as function of time.

Explanation

Heat produced by coil inside the cylinder in time dt is
`dQ=l^(2)rdt………………(i)`
(a) As we know
`rArrdQ=nC_(P)dT`
`I^(2)rdt=(5R)/2dT(C_(P)=(5R)/2)`
Here `T=T_(0)+"at"+bt^(2)rArr(dT)/(dt)=(a+2bt)`
`I=sqrt([(5R)/(2r)(2bt+a)])............(ii)`
`(C_(p)=(5R)/2)`
Pressure is constant `PV=RTrArrPdV=RdT`
PAdx=RdT
`"Velocit v"=(dx)/(dt)=R/(PA),(dT)/(dt)=R/(PA)(2bt+a)("here"P_(0)A+mg=PA)`
`=R/(P_(0)A+mg)(2bt+a)`