Ice at \( -20^{\circ} \mathrm{C} \) is filled upto height \( \mathrm{h}=10 \mathrm{~cm} \) in a uniform cylindrical vessel. Water at temperature \( \theta^{\circ} \mathrm{C} \) is filled in another identical vessel upto the same height \( \mathrm{h}=10 \mathrm{~cm} \). Now, water from second vessel is poured into first vessel and it is found that level of upper surface falls through \( \Delta \mathrm{h}=0.5 \mathrm{~cm} \) when thermal equilibrium is reached. Neglecting thermal capacity of vessels, change in density of water due to change in temperature and loss of heat due to radiation, calculate initial temperature \( \theta \) of water.
Given,
Density of water: \( \quad \rho_{\mathrm{w}}=1 \mathrm{gm} \mathrm{cm}^{-3} \quad \) Density of ice : \( \rho_{\mathrm{i}}=0.9 \mathrm{gm} / \mathrm{cm}^{3} \)
Specific heat of water: \( \mathrm{s}_{\mathrm{w}}=1 \mathrm{cal} / \mathrm{gm}^{0} \mathrm{C} \quad \) Specific heat of ice : \( \quad \mathrm{s}_{\mathrm{i}}=0.5 \mathrm{cal} / \mathrm{gm}^{\circ} \mathrm{C} \)
Specific latent heat of ice : \( \mathrm{L}=80 \mathrm{cal} / \mathrm{gm} \)
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