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A copper ring has a diameter of exactly $25 \,\,mm$ at its temperature of $0^o C$. An aluminium sphere has a diameter of exactly $25.05\,\, mm$ at its temperature of $100^o C$. 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 : $\alpha_{Cu} = 17 × 10^{-6} /^o C, \alpha = 2.3 × 10^{-5} /^o C,$ specific heat of $Cu = 0.0923 Cal/g^o C$ and specific heat of $Al = 0.215 cal/g^o C$)
$1/5$
$23/108$
$23/54$
$216/23$
Solution
A equilibrium $(M s \Delta t)_{C u}=(M s \Delta t)_{A l}$
When sphere passing through ring then Radius of sphere $=$ Radius ring
$\left(l_{2}\right)_{C u}=\left(l_{2}\right)_{A l}$
$25\left(1+\alpha_{C u} \Delta t\right)=25.05\left(1+\alpha_{A l} \Delta t\right)$
