A cylindrical block of length 0.4 m and area of cross-section `0.04 m^2` is placed coaxially on a thin metal disc of mass 0.4 kg and of the same cross - section. The upper face of the cylinder is maintained at a constant temperature of 400 K and the initial temperature of the disc is 300K. if the thermal conductivity of the material of the cylinder is `10 "watt"// m.K ` and the specific heat of the material of the disc is `600J//kg.K`, how long will it take for the temperature of the disc to increase to 350 K? Assume for purpose of calculation the thermal conductivity of the disc to be very high and the system to be thermally insulated except for the upper face of the cylinder.

Explanation

In steady state, fundamental equation of heat conduction is
`H=(dQ)/(dt)=(kA(T_(1)-T_(2)))/l`
Here temperature of upper face of cylinder is 400 K and that of lower face is changing
`(dQ)/(dt)=kA((400-T))/l`
Where T is intermediate temperature of lower face of cylinder dQ is heat absorbed by disc in time dt.
Also `dQ=mcdT`
where m=mass of disc =0.4 kg ,c=specific heat of disc
`=600` J/kg .K
`(mCdT)/(dt)=kA((400-T))/l`
`impliesint_(0)^(1)dt=(mcl)/(kA)int_(300)^(350)=(dT)/(400-T)`
`t=(mcl)/(kA)[("In"(400-T))/(-1)]_(300)^(350)=(mc l)/(kA)"In" ((400-300)/(400-350))`
Substituting the value of m,c,l, k and A we get
`t=166.3` sec