Question

An ice ball has a metal piece embedded into it. The temperature of the ball is - θ ∘ C and it contains mass M of ice. When placed in a large tub containing water at 0 ∘ C , it sinks. Assume that the water in immediate contact with the ice ball freezes and thereby size of the ball grows. What is the maximum possible mass of the metal piece so that the ball can eventually begin to float. Densities of ice, water and metal are σ , ρ and d respectively. Specific heat capacity of ice is s and its specific latent heat is L. Neglect the heat capacity of the metal piece.

Answer 1

M(1 + \frac{s θ}{L}) \frac{d(ρ - σ)}{σ(d - ρ)}

Answer 2

1. Calculate the heat required to raise the initial mass of ice from $-\theta^\circ C$ to $0^\circ C$.

2. This heat is supplied by the freezing of water at $0^\circ C$. Calculate the mass of water that freezes using the latent heat of fusion.

3. The frozen water adds to the mass of ice. Calculate the new total mass of ice.

4. Calculate the total mass of the ball (new ice mass + metal mass).

5. Calculate the total volume of the ball (volume of new ice mass + volume of metal mass).

6. For the ball to float, its average density must be equal to the density of water. Set the total mass divided by the total volume equal to the density of water.

7. Solve the resulting equation for the mass of the metal piece. This gives the maximum mass of the metal piece for which the ball can eventually float.