A cubical block of wood of specific gravity `0.5` and chunk of concrete of specific gravity `2.5` are fastened together. the ratio of mass of wood to the mass of concrete which makes the combination to float with entire volume of the combination submerged in water is

Options

A. `(5)/(3)`
B. `(6)/(5)`
C. `(5)/(2)`
D. `(4)/(3)`

Answer

To solve the problem, we need to find the ratio of the mass of wood to the mass of concrete when they are fastened together and completely submerged in water.

### Step-by-Step Solution:

1. **Understanding Specific Gravity**:
- Specific gravity (SG) is defined as the ratio of the density of a substance to the density of water.
- For wood, SG = 0.5, which means its density \( \rho_{wood} = 0.5 \times \rho_{water} \).
- For concrete, SG = 2.5, which means its density \( \rho_{concrete} = 2.5 \times \rho_{water} \).

2. **Let Volumes be Defined**:
- Let the volume of wood be \( V_1 \) and the volume of concrete be \( V_2 \).

3. **Weight of the System**:
- The total weight of the system (wood + concrete) is given by:
\[ W = m_{wood} + m_{concrete} = \rho_{wood} \cdot V_1 + \rho_{concrete} \cdot V_2 \]

4. **Buoyant Force**:
- The buoyant force acting on the submerged object is equal to the weight of the water displaced by the total volume \( V_1 + V_2 \):
\[ F_b = \rho_{water} \cdot g \cdot (V_1 + V_2) \]

5. **Equating Weight and Buoyant Force**:
- For the combination to float with the entire volume submerged, the weight of the system must equal the buoyant force:
\[ \rho_{wood} \cdot V_1 + \rho_{concrete} \cdot V_2 = \rho_{water} \cdot (V_1 + V_2) \]

6. **Substituting Densities**:
- Substitute the densities in terms of specific gravity:
\[ (0.5 \cdot \rho_{water}) \cdot V_1 + (2.5 \cdot \rho_{water}) \cdot V_2 = \rho_{water} \cdot (V_1 + V_2) \]
- Dividing through by \( \rho_{water} \):
\[ 0.5 V_1 + 2.5 V_2 = V_1 + V_2 \]

7. **Rearranging the Equation**:
- Rearranging gives us:
\[ 0.5 V_1 + 2.5 V_2 - V_1 - V_2 = 0 \]
\[ -0.5 V_1 + 1.5 V_2 = 0 \]
\[ 1.5 V_2 = 0.5 V_1 \]
\[ \frac{V_1}{V_2} = \frac{1.5}{0.5} = 3 \]

8. **Finding the Mass Ratio**:
- The mass of wood \( m_{wood} = \rho_{wood} \cdot V_1 = (0.5 \cdot \rho_{water}) \cdot V_1 \)
- The mass of concrete \( m_{concrete} = \rho_{concrete} \cdot V_2 = (2.5 \cdot \rho_{water}) \cdot V_2 \)
- The ratio of mass of wood to mass of concrete is:
\[ \frac{m_{wood}}{m_{concrete}} = \frac{(0.5 \cdot \rho_{water}) \cdot V_1}{(2.5 \cdot \rho_{water}) \cdot V_2} = \frac{0.5 \cdot V_1}{2.5 \cdot V_2} \]
- Substituting \( \frac{V_1}{V_2} = 3 \):
\[ \frac{m_{wood}}{m_{concrete}} = \frac{0.5 \cdot 3}{2.5} = \frac{1.5}{2.5} = \frac{3}{5} \]

### Final Answer:
The ratio of the mass of wood to the mass of concrete is \( \frac{3}{5} \).