A small blackened copper sphere of mass and specific heat capacity

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Mathematical Methods for Physics, Ch First-order differential equations and physical modelling

Problem

A small blackened copper sphere of mass m=0.050 kgm = 0.050\,\mathrm{kg} and specific heat capacity c=385 J kg−1 K−1c = 385\,\mathrm{J\,kg^{-1}\,K^{-1}}, of surface area As=2.0×10−3 m2A_s = 2.0\times10^{-3}\,\mathrm{m^2}, is heated to T0=400 KT_0 = 400\,\mathrm{K} and then left to cool in surroundings at Ta=300 KT_a = 300\,\mathrm{K}. Two models are proposed for the power it loses. Model A (Newton's law of cooling, valid when convection in the surrounding air dominates): Ploss=hAs(T−Ta)P_{\text{loss}} = hA_s(T - T_a), with heat-transfer coefficient h=15 W m−2 K−1h = 15\,\mathrm{W\,m^{-2}\,K^{-1}}. Model B (pure radiative loss, valid in a vacuum): Ploss=ϵσAs(T4−Ta4)P_{\text{loss}} = \epsilon\sigma A_s(T^4 - T_a^4), with emissivity ϵ=0.80\epsilon = 0.80 and Stefan-Boltzmann constant σ=5.670×10−8 W m−2 K−4\sigma = 5.670\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}. In both cases energy conservation gives mc T˙=−Plossmc\,\dot T = -P_{\text{loss}}. ((a)) Write the differential equation for T(t)T(t) under each model, and state the order and linearity of each. ((b)) Compute the initial rate of cooling T˙\dot T at t=0t=0 under Model A, in K s−1\mathrm{K\,s^{-1}}. ((c)) Compute the initial rate of cooling T˙\dot T at t=0t=0 under Model B, in K s−1\mathrm{K\,s^{-1}}. ((d)) Say which part of mcT˙=−Plossmc\dot T = -P_{\text{loss}} is fixed by the definition of heat capacity and energy conservation, and which part is the modeller's choice, and explain in one sentence why the two models disagree even though both conserve energy correctly.

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