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))`.

A uniform cylinder of length `L` and thermal conductivity `k` is place

1.	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» Correct Answer - `(mCL)/(KS)ln((T_(0)-T_(1))/(T_(0)-T_(2)))` 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))\|`

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