A cubical block of ice floating in water has to suport metal piece weighing 0.5 kg. What can be the minimum edge of the block so that it does not sink in water/ specific gravity of ice=0.9.

Explanation

To solve the problem of determining the minimum edge length of a cubical block of ice that can float while supporting a metal piece weighing 0.5 kg, we can follow these steps:

### Step 1: Understand the Forces Acting on the Block
The block of ice must balance the weight of the metal piece and its own weight with the buoyant force acting on it. The forces can be expressed as:
- Weight of the metal piece (W_metal) = m_metal * g
- Weight of the ice block (W_ice) = m_ice * g
- Buoyant force (F_buoyant) = ρ_water * V_submerged * g

Where:
- m_metal = 0.5 kg (weight of the metal piece)
- ρ_water = 1000 kg/m³ (density of water)
- V_submerged = volume of the submerged part of the ice block

### Step 2: Relate the Mass of Ice to its Volume
The specific gravity of ice is given as 0.9, which means:
- ρ_ice = 0.9 * ρ_water = 0.9 * 1000 kg/m³ = 900 kg/m³

The volume of the ice block (V_ice) can be expressed in terms of its edge length (x):
- V_ice = x³

The mass of the ice block can then be calculated as:
- m_ice = ρ_ice * V_ice = 900 kg/m³ * x³

### Step 3: Set Up the Equation for Equilibrium
The total weight supported by the buoyant force can be expressed as:
- F_buoyant = W_ice + W_metal

Substituting the expressions for weight:
- ρ_water * V_submerged * g = (900 kg/m³ * x³ * g) + (0.5 kg * g)

Since g cancels out from both sides, we have:
- 1000 kg/m³ * V_submerged = 900 kg/m³ * x³ + 0.5 kg

### Step 4: Determine the Volume Submerged
For a floating block, the volume submerged (V_submerged) is equal to the volume of the ice block:
- V_submerged = x³

Substituting this into the equation gives:
- 1000 kg/m³ * x³ = 900 kg/m³ * x³ + 0.5 kg

### Step 5: Rearranging the Equation
Rearranging the equation to isolate x³:
- 1000 x³ - 900 x³ = 0.5
- 100 x³ = 0.5
- x³ = 0.5 / 100
- x³ = 0.005 m³

### Step 6: Calculate the Edge Length
To find the edge length x, take the cube root:
- x = (0.005)^(1/3)

Calculating this gives:
- x ≈ 0.17 m or 17 cm

### Conclusion
The minimum edge length of the block of ice that can float while supporting the metal piece is approximately **17 cm**.

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