# Comparing rates of cooling

There are 2 metallic spheres A and B. Mass of A is thrice the mass of B. Both the spheres are heated to the same high temperature from same initial temperature. The spheres are thermally insulated from each other. I have to find the rate of cooling, so I tried Newton's Law of Cooling, but that doesn't seem to take in consideration the relation between the masses, and yields the wrong answer (is NLC valid for high temperatures?). I thought of the formula u = eAσT^4, but I'm not sure how to use this. Please suggest an approach.

• 'u = eAσT^4' doesn't have much to do with NLC (which is valid for high temperatures with caveats) Revisit NLC here: en.wikipedia.org/wiki/… and get inspired again. – Gert Nov 26 '15 at 3:48

Newton's law of cooling is a corollary of Fourier's law of heat conduction:

$$q=-\kappa \nabla T,$$

where $q$ is the heat flux, $\kappa$ the heat conductivity and $\nabla T$ the temperature gradient (in a single dimension $\nabla T=\frac{dT}{dx}$). In essence this law tells us that heat flows from hot to cold and that the heat flow is proportional to the spatial temperature gradient.

Reworked for a body at temperature $T(t)$, cooling in a colder environment at constant temperature $T_0$ we get Newton's law of cooling:

$$\frac{dQ}{dt}=-hA[T(t)-T_0],$$

with $Q$ the heat energy of the object, $h$ the heat transfer coefficient and $A$ the total surface area of the object.

When the object loses an infinitesimal amount of heat energy $dQ$ it also drops in temperature a bit:

$$dQ=mc_pdT,$$

where $m$ is the mass of the object and $c_p$ the specific heat capacity of the object.

Substituting into the first equation we get:

$$mc_p\frac{dT}{dt}=-hA[T(t)-T_0],$$

Integrated between $0,T_1$ and $t,T_2$ we get:

$$\ln\frac{T_2-T_0}{T_1-T_0}=-\frac{hA}{mc_p}t,$$

$$\frac{T_2-T_0}{T_1-T_0}=e^{-\alpha t},$$

with:

$$\alpha=\frac{hA}{mc_p}$$

There are 2 metallic spheres A and B. Mass of A is thrice the mass of B.

Your task is now to work out the influence of $m$ on $\alpha$, bearing in mind that $m$ also has an effect on $A$. In the case of two spheres with $m_A=3m_B$ it should be easy to work out the ration of $\alpha_A$ and $\alpha_B$, which gives you the ratio of the cooling rate of the two spheres.

Thermally insulated tells you go for Stefan. Newton's law of cooling is for convection transfer. Don't confuse stuff. Maybe you missed the term "made of the same material", you should conclude the relation in their surface area.