A block B of specific gravity 2 and another block C of specific gravity `0.5` .Both are joined together and they are floating water such that they are completely dipped inside water , the ratio of the masses of the blocks B and C is

Options

A. `2:1`
B. `3:2`
C. `5:3`
D. `2:3`

Answer

To find the ratio of the masses of blocks B and C, we can follow these steps:

### Step 1: Understand the Specific Gravity
Specific gravity is defined as the ratio of the density of a substance to the density of water.
- For block B (specific gravity = 2), the density \( \rho_B = 2 \cdot \rho_w \)
- For block C (specific gravity = 0.5), the density \( \rho_C = 0.5 \cdot \rho_w \)

### Step 2: Apply the Floating Condition
When the blocks are floating, the total weight of the blocks must equal the buoyant force acting on them. The buoyant force is equal to the weight of the water displaced, which can be expressed as:
\[ \text{Buoyant Force} = \rho_w \cdot g \cdot (V_B + V_C) \]
where \( V_B \) and \( V_C \) are the volumes of blocks B and C, respectively.

### Step 3: Relate Volume to Mass
The volume of each block can be expressed in terms of mass and density:
\[ V_B = \frac{m_B}{\rho_B} \quad \text{and} \quad V_C = \frac{m_C}{\rho_C} \]

### Step 4: Set Up the Equation
From the floating condition, we have:
\[ m_B g + m_C g = \rho_w g (V_B + V_C) \]
Cancelling \( g \) from both sides gives:
\[ m_B + m_C = \rho_w \left( \frac{m_B}{\rho_B} + \frac{m_C}{\rho_C} \right) \]

### Step 5: Substitute Densities
Substituting the expressions for \( \rho_B \) and \( \rho_C \):
\[ m_B + m_C = \rho_w \left( \frac{m_B}{2 \rho_w} + \frac{m_C}{0.5 \rho_w} \right) \]
This simplifies to:
\[ m_B + m_C = \frac{m_B}{2} + 2m_C \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ m_B + m_C - \frac{m_B}{2} - 2m_C = 0 \]
This can be rewritten as:
\[ \frac{1}{2} m_B - m_C = 0 \]
or
\[ \frac{1}{2} m_B = m_C \]

### Step 7: Finding the Ratio
From the equation \( m_C = \frac{1}{2} m_B \), we can express the ratio of the masses:
\[ \frac{m_B}{m_C} = 2 \]

### Final Answer
Thus, the ratio of the masses of block B to block C is:
\[ \frac{m_B}{m_C} = 2:1 \]