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 solve the problem of finding the ratio of the masses of block B to block C, we can follow these steps:

### Step 1: Understand the Problem
We have two blocks:
- Block B with a specific gravity of 2.
- Block C with a specific gravity of 0.5.

Both blocks are completely submerged in water.

### Step 2: Define the Masses
Let:
- \( m_B \) = mass of block B
- \( m_C \) = mass of block C

### Step 3: Understand Specific Gravity
Specific gravity (SG) is the ratio of the density of a substance to the density of water. Therefore:
- Density of block B, \( \rho_B = 2 \cdot \rho_w \)
- Density of block C, \( \rho_C = 0.5 \cdot \rho_w \)

Where \( \rho_w \) is the density of water.

### Step 4: Calculate the Volume of Each Block
The volume of each block can be expressed in terms of mass and density:
- Volume of block B, \( V_B = \frac{m_B}{\rho_B} = \frac{m_B}{2 \cdot \rho_w} \)
- Volume of block C, \( V_C = \frac{m_C}{\rho_C} = \frac{m_C}{0.5 \cdot \rho_w} \)

### Step 5: Apply the Principle of Buoyancy
According to Archimedes' principle, the buoyant force acting on the blocks is equal to the weight of the water displaced by them:
- Weight of the blocks (downward force): \( W = m_B g + m_C g \)
- Buoyant force (upward force): \( F_B = \rho_w (V_B + V_C) g \)

Setting these equal gives us:
\[ m_B g + m_C g = \rho_w (V_B + V_C) g \]

### Step 6: Simplify the Equation
Cancel \( g \) from both sides:
\[ m_B + m_C = \rho_w \left( \frac{m_B}{2 \cdot \rho_w} + \frac{m_C}{0.5 \cdot \rho_w} \right) \]

This simplifies to:
\[ m_B + m_C = \frac{m_B}{2} + 2m_C \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ m_B + m_C - \frac{m_B}{2} - 2m_C = 0 \]
\[ \frac{m_B}{2} - m_C = 0 \]

### Step 8: Solve for the Ratio
From the equation:
\[ \frac{m_B}{2} = m_C \implies m_B = 2m_C \]

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

### Final Answer
The ratio of the masses of block B to block C is \( 2:1 \).

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