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What is the effect of specific heat capacity on rate of cooling?

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I was wondering how two BODIES MADE of the same material and having the same Surface AREA but DIFFERENT Volumes, heated to the same temperature, would cool. Would they cool at the same rate? I know that Stefan's Law states that E=σAT4E=σAT4 so THEORETICALLY they should radiate the same amount of energy per unit time and hence cool at the same rate. But doesn't this violate the conservation of energy? How could two bodies having different thermal energies(due to different volumes and masses) have the same rate of cooling?



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