Question

Some pieces of impurity (density =$\rho$) is embedded in ice. This ice is floating in water. (density =$\rho_w$). When ice melts, level of water will

• A. fall if $\rho > \rho_w$
• B. fall if $\rho <\rho_w$
• C. remain unchanged, if $\rho < \rho_w$
• D. rise if $\rho > \rho_w$

Answer 1

Answer: (A)

fall if $\rho > \rho_w$

Answer 2

Let $m_{ice}$ be the mass of pure ice and $m_{imp}$ be the mass of impurity. Let $V_{ice}$ be the volume of pure ice and $V_{imp}$ be the volume of impurity. The total mass of the ice block is $M = m_{ice} + m_{imp}$. When floating, the volume of displaced water is $V_{submerged} = \frac{M}{\rho_w} = \frac{m_{ice} + m_{imp}}{\rho_w}$. When melted, the ice forms water of volume $V_{melted\_ice} = \frac{m_{ice}}{\rho_w}$. The impurity occupies volume $V_{imp\_final} = \frac{m_{imp}}{\rho}$. The total volume after melting is $V_{final} = \frac{m_{ice}}{\rho_w} + \frac{m_{imp}}{\rho}$. The change in volume is $\Delta V = V_{final} - V_{submerged} = \left(\frac{m_{ice}}{\rho_w} + \frac{m_{imp}}{\rho}\right) - \frac{m_{ice} + m_{imp}}{\rho_w} = m_{imp} \left(\frac{1}{\rho} - \frac{1}{\rho_w}\right) = m_{imp} \frac{\rho_w - \rho}{\rho \rho_w}$. If $\rho > \rho_w$, then $\Delta V < 0$, so the water level falls. If $\rho < \rho_w$, then $\Delta V > 0$, so the water level rises. If $\rho = \rho_w$, then $\Delta V = 0$, so the water level remains unchanged.