Two sheres made of same material have their radii in the ratio `1:3` They are heated to the same temperature and kept in the same surroudings at a moderate temperature Show that the ratio of their initial rates of fall of temperature is `3:1` if the bodies are cooled by natural convection and radiation .

Explanation

The rate of fall to temperature of a hot body is given by
`(d theta)/(dt) = (eAsigma)/(ms) (T^(4) -T_(0)^(4))`
`:.(d"theta)/(dt) = (e4 pi R^(2) sigma(T^(4)-T_(0)^(2)))/((4)/(3) pi R^(3) rhos)rArr(d theta)/(dt) prop(1)/(R )`
`((d theta//dt)_1)/((d theta//dt)_(2)) = (R_(2))/(R_(1)) = (3)/(1) ( :.(R_(1))/(R_(2) )=1:3)`
Their ratio of fall of temperature is `3:1` .