Newtons Law dTdt kTt Tm where Tt is the temperature of an

Newton\'s Law dT/dt = k(T(t) - T_m) (where T(t) is the temperature of an object at time t, and T_m is the temperature of the surrounding medium) is both a Law of Cooling and a Law of Heating. How can this be? Does it depend on the value of k? Discuss.

Solution

This Newton equation provides you with the value of temperature of a body kept in medium as the time progress. Saying that, it means it provides you with Temperature as a function of time. This dT/dt varies linearly with _T. i.e. difference in the temperature of the body and the medium (T_b(t)-T_m).

T_b(t) : T_body at time t

T_m: : Temperature of ambience/medium

So it could be possible that medium is cooler than the body (e.g. Hot soup kept at room temperature) or body is cooler than the medium (Ice cube kept at room temperature).

dT/dt is directly proportional to T(t)-T_m where T(t) temperature at time t and T_m is Temperature of medium

dT/dt = K (T(t)-T_m) where K is constant and K>0,

Saying that T(t)-T_m could be positive or could be negative depending on the situation (Hot soup or Ice cube)

When T(t)-T_m is positive it means object (Hot soup) is cooling down to the surrounding temperature. Since the body is cooling down it means the that the derivation dT/dt is negative, temperature of the body is decreasing as the time progress

Hence the equation becomes dT/dt = -K (T(t)-T_m)

When T(t)-T_m is negative it means object (Ice cube) is warming up to the surrounding temperature. Since the body is warming up to ambience temperature it means the that the derivation dT/dt is positive, temperature of the body is increasing as the time progress

Hence the equation becomes dT/dt = K (T(t)-T_m)

Hence the same equation is valid for both heating or cooling.

Second Part:

K is a proportionality constant and does not depend on Heating/cooling.

General Solution of the above differential equation leads to following general solution

T(t) - T_m= (T₀ - T_m)e^-kt where T₀: temperature of body at time t=0, T(t): temperature of body at time t ,T_m: Ambience temp.

Say if we know T_0, T_m and we also know that say at time t= 10 min we now the temperature of body. This will help us find the value of k from above equation.

Now once we know this K, we can determine at further time t=20 min or 5 hours what will be the temperature of body. We need not to measure it.