A cubical block of ice floating in water has to support a 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, we need to determine the minimum edge length of a cubical block of ice that can float while supporting a metal piece weighing 0.5 kg. The specific gravity of ice is given as 0.9.

### Step-by-Step Solution:

1. **Understand the Forces Involved**:
- The block of ice is floating in water and must support the weight of the metal piece.
- The buoyant force acting on the ice must equal the total weight of the ice plus the weight of the metal piece.

2. **Identify the Weights**:
- The weight of the metal piece (W_m) is given by:
\[ W_m = m \cdot g = 0.5 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 4.9 \, \text{N} \]
- The weight of the ice block (W_i) can be expressed as:
\[ W_i = \text{density of ice} \cdot g \cdot V_i \]
- The volume of the ice block (V_i) is \(x^3\) where \(x\) is the edge length of the cube.

3. **Calculate the Buoyant Force**:
- The buoyant force (B) acting on the ice is equal to the weight of the water displaced, which is:
\[ B = \text{density of water} \cdot g \cdot V_d \]
- Since the volume of the ice block submerged is equal to its volume when floating, we have:
\[ V_d = x^3 \]
- Therefore, the buoyant force can be expressed as:
\[ B = \text{density of water} \cdot g \cdot x^3 \]

4. **Set Up the Equation for Equilibrium**:
- At equilibrium, the buoyant force equals the total weight:
\[ B = W_m + W_i \]
- Substituting the expressions we have:
\[ \text{density of water} \cdot g \cdot x^3 = W_m + \text{density of ice} \cdot g \cdot x^3 \]

5. **Substituting Known Values**:
- The density of water is \(1 \, \text{g/cm}^3\) and the specific gravity of ice is \(0.9\), so:
\[ \text{density of ice} = 0.9 \, \text{g/cm}^3 \]
- The equation simplifies to:
\[ 1 \cdot g \cdot x^3 = 4.9 + 0.9 \cdot g \cdot x^3 \]

6. **Cancel \(g\) and Rearrange**:
- Cancel \(g\) from both sides:
\[ x^3 = 4.9 + 0.9 x^3 \]
- Rearranging gives:
\[ x^3 - 0.9 x^3 = 4.9 \]
- This simplifies to:
\[ 0.1 x^3 = 4.9 \]
- Thus:
\[ x^3 = \frac{4.9}{0.1} = 49 \]

7. **Calculate the Edge Length**:
- Taking the cube root:
\[ x = \sqrt[3]{49} \approx 3.65 \, \text{cm} \]

### Final Answer:
The minimum edge length of the block of ice required to support the metal piece without sinking is approximately **3.65 cm**.