A copper ring has a diameter of exactly 25 mm at its temperature of 0°C. An aluminium sphere has a diameter of exactly 25.05 mm at its temperature of 100°C. The sphere is placed on top of the ring and two are allowed to come to thermal equilibrium, no heat being lost to the surrounding. The sphere just passes through the ring at the equilibrium temperature. The ratio of the mass of the sphere & ring is: (given : αCu = 17 × 10 – 6 /°C, αAl = 2.3 × 10 – 5 /°C, specific heat of Cu = 0.0923 Cal/g°C and specific heat of Al = 0.215 cal/g°C)
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At thermal equilibrium, the aluminum sphere just passes through the copper ring, meaning their diameters are equal. Let the equilibrium temperature be T. The final diameter for both is the same. The initial diameter of the copper ring at 0°C is 25 mm, so its diameter at T is: DCu = 25[1 + αCu(T - 0)]. The initial diameter of the aluminum sphere at 100°C is 25.05 mm, so its diameter at T is: DAl = 25.05[1 + αAl(T - 100)]. Setting them equal: 25(1 + 17×10-6T) = 25.05[1 + 23×10-6(T - 100)]. Solving this gives T = 44.4°C.
Heat lost by sphere = Heat gained by ring. mAlcAl(100 - T) = mCucCu(T - 0). Substituting values: mAl×0.215×(55.6) = mCu×0.0923×(44.4). The ratio msphere/mring = mAl/mCu = (0.0923×44.4)/(0.215×55.6) = 23/108.
Final Answer: 23/108