A sphere and a cube of same material and same total surface area are placed in the same evacuated space turn by turn after they are heated to the same temperature. Find the ratio of their initial rates of cooling in the enclosure.

Question

Infinity Learn · Accepted Answer

Rate of emission of energy $$=$$ σ$$T4sLet$$ $$m1$$ be the mass of sphere, C is specific heat and (d$$θ$$dt), the rate of cooling and s is surface areaFor sphere           σ$$T4S$$ $$=$$ $$m1C(d$$θ$$dt)--------(i)Let$$ $$m2$$ be the mass of cube, C its specific heat and (d$$θ$$dt), the rate of coolingFor cube σ$$T4S$$ $$=$$ $$m2C(d$$θ$$dt)------(ii)From$$ equations (i) and (ii)(d$$θ$$dt)s(d$$θ$$dt)c1$$ $$=$$ $$m2m1=$$ RsRcor $$a3$$ρ$$(43)π$$r2$$ρ $$=$$ RsRcwhere a is the side of cube and r is the radius of sphere, ρ is the density∴ RsRc $$=$$ $$3a34$$$$π$$$$r3But$$ since S is the same, $$so6a2$$ $$=$$ $$4$$$$π$$$$r2or$$ $$a2$$ $$=$$ (23)π$$r2$$∴ RsRc $$=$$ $$3(2$$$$π$$$$r2/3)324$$$$π$$$$r3$$ $$=$$ $$2$$$$π$$$$2$$$$π$$$$3(4$$π$$)=$$ $$2$$$$π$$$$12$$ $$=$$ π$$6$$

A sphere and a cube of same material and same total surface area are placed in the same evacuated space turn by turn after they are heated to the same temperature. Find the ratio of their initial rates of cooling in the enclosure.

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