Two spheres made of same material have their radii in the ratio 1 : 3. They are heated lo the same temperature and kept in the same cool surroundings at a moderate temperature. Show that the ratio of their initial rates off all of temperature is 3 : 1 if the bodies are cooled by natural convection and radiation

Explanation

The rate of fall of temperature of a hot body is given by `(d theta)/(dt ) = (e A sigma)/(ms ) (T^(4) - T_(0)^(4) ) `
For a spherical body, A = 4pi`r^(2) & m = (4)/(3) pi r^(3) rho `
`therefore (d theta )/(dt ) = (e.4 pi r^(2) sigma (T^(4) - T_(0)^(4)))/((4)/(3) pi r^(3) rho s) rArr (d theta)/(dt ) alpha (1)/(r)`
`therefore ((d theta_(1)//dt ))/((d theta_(2)//dt )) = (r_(2))/(r_(1)) = (3)/(1)`