The ratio of masses of a cubical block of wood and a chunk of concrete is 3/5 so that the combination just floats with entire volume submerged under water. If the specific gravity of wood is 0.5 then specific gravity of concrete will be

Options

A. 2.5
B. 25
C. 15
D. 3

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let:
- Mass of wood = \(3x\)
- Mass of concrete = \(5x\)
- Density of wood = \(\rho_1\)
- Density of concrete = \(\rho_2\)
- Density of water = \(\rho_w\)

### Step 2: Write the buoyancy condition
Since the combination of wood and concrete just floats with the entire volume submerged, we can use the principle of buoyancy:
\[ \rho_w \cdot V_{\text{block}} = \text{Weight of the block} \]
This means:
\[ \rho_w \cdot V_{\text{block}} = \text{mass of wood} + \text{mass of concrete} \]

### Step 3: Express the volume of the block
The volume of the block can be expressed as:
\[ V_{\text{block}} = V_{\text{wood}} + V_{\text{concrete}} = \frac{3x}{\rho_1} + \frac{5x}{\rho_2} \]
The total mass of the block is:
\[ \text{mass of block} = 3x + 5x = 8x \]

### Step 4: Set up the equation
From the buoyancy condition, we have:
\[ \rho_w \left( \frac{3x}{\rho_1} + \frac{5x}{\rho_2} \right) = 8x \]
We can cancel \(x\) from both sides (assuming \(x \neq 0\)):
\[ \rho_w \left( \frac{3}{\rho_1} + \frac{5}{\rho_2} \right) = 8 \]

### Step 5: Substitute the specific gravity of wood
The specific gravity of wood is given as 0.5, which means:
\[ \frac{\rho_1}{\rho_w} = 0.5 \implies \rho_1 = 0.5 \rho_w \]

### Step 6: Substitute \(\rho_1\) into the equation
Substituting \(\rho_1\) into the equation gives:
\[ \rho_w \left( \frac{3}{0.5 \rho_w} + \frac{5}{\rho_2} \right) = 8 \]
This simplifies to:
\[ \rho_w \left( 6 + \frac{5}{\rho_2} \right) = 8 \]

### Step 7: Divide by \(\rho_w\)
Dividing both sides by \(\rho_w\) gives:
\[ 6 + \frac{5}{\rho_2} = \frac{8}{\rho_w} \]

### Step 8: Rearrange to find \(\rho_2\)
Rearranging gives:
\[ \frac{5}{\rho_2} = \frac{8}{\rho_w} - 6 \]
\[ \frac{5}{\rho_2} = \frac{8 - 6\rho_w}{\rho_w} \]
Thus,
\[ \rho_2 = \frac{5 \rho_w}{8 - 6\rho_w} \]

### Step 9: Find the specific gravity of concrete
The specific gravity of concrete is given by:
\[ \text{Specific Gravity of Concrete} = \frac{\rho_2}{\rho_w} = \frac{5}{8 - 6\rho_w} \]
Substituting \(\rho_w = 1\) (as the specific gravity of water is 1):
\[ \text{Specific Gravity of Concrete} = \frac{5}{8 - 6} = \frac{5}{2} = 2.5 \]

### Final Answer
The specific gravity of concrete is \(2.5\).

---