A piece of ice is floating in a glass vessel filled with water. Then prove that level of water in the vessel remains unchanged after melting of ice.

Explanation

To prove that the level of water in the vessel remains unchanged after the melting of ice, we can follow these steps:

### Step 1: Understand the situation
We have a piece of ice floating in water. The ice is partially submerged, and the water level in the vessel is at a certain height.

### Step 2: Apply Archimedes' Principle
According to Archimedes' Principle, the buoyant force acting on the ice is equal to the weight of the water displaced by the submerged part of the ice.

Let:
- \( m_d \) = mass of the ice
- \( V_i \) = volume of the ice
- \( \rho_w \) = density of water
- \( g \) = acceleration due to gravity

### Step 3: Calculate the buoyant force
The buoyant force \( F_b \) can be expressed as:
\[ F_b = V_d \cdot \rho_w \cdot g \]
where \( V_d \) is the volume of water displaced by the submerged part of the ice.

Since the ice is floating, the buoyant force equals the weight of the ice:
\[ F_b = m_d \cdot g \]

### Step 4: Relate the volume of ice to the volume of displaced water
The volume of ice that is submerged can be calculated as:
\[ V_d = \frac{m_d}{\rho_w} \]
This volume \( V_d \) is equal to the volume of water displaced by the ice.

### Step 5: Consider the melting of ice
When the ice melts, it turns into water. The mass of the melted ice is the same as the mass of the ice before it melted, \( m_d \). The volume of the melted ice (now water) can be expressed as:
\[ V_m = \frac{m_d}{\rho_{ice}} \]
where \( \rho_{ice} \) is the density of ice.

### Step 6: Compare volumes
Since the density of ice is less than that of water, the volume of water produced by the melted ice will be equal to the volume of water that was displaced when the ice was floating:
\[ V_d = V_m \]

### Step 7: Conclusion
Since the volume of water produced by the melted ice is equal to the volume of water displaced by the ice while it was floating, the overall water level in the vessel remains unchanged after the ice melts.

### Final Statement
Thus, we can conclude that the level of water in the vessel remains unchanged after the melting of the ice.

---