A uniform cylinder of length `L` and thermal conductivity `k` is placed on a metal plate of the same area `S` of mass `m` and infinite conductivity. The specific heat of the plate is `c`. The top of the cylinder is maintained at `T_(0)`. Find the time required for the temperature of the plate to rise from `T_(1)` to `T_(2)(T_(1) lt T_(2) lt T_(0))`.

Answer

`(mCL)/(KS)ln((T_(0)-T_(1))/(T_(0)-T_(2)))`

Explanation

The whole metal plate will always be at uniform temperature `(T)` since it has infinite conductivity.
then, `-mc (dT)/(dt) = (ks(T_(0)-T))/(l)`
`rArr -mc overset(T_(2))underset(T_(1))int (dt)/(T_(0)-T) = (ks)/(l) overset(t)underset(0)int dt`
`ln ((T_(0)-T_(2))/(T_(0)-T_(1))) = (ks t)/(mcl)`
`rArr t = (mcl)/(ks) ln |(T_(0)-T_(2))/(T_(0)-T_(1))|`