A body of density d is counterpoised by Mg of weights of density `d_1` in air of density d. Then the true mass of the body is

Options

A. M
B. `M(1-d/d_2)`
C. `M(1-d/d_1)`
D. `M((1-d/d_2)/(1-d/d_1))`

Answer

To solve the problem step by step, we need to analyze the forces acting on the body when it is submerged in air and counterpoised by another mass.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a body with density \( d \) and mass \( m_0 \).
- This body is counterpoised by another mass \( m \) of density \( d_1 \) in air, which has a density \( d \).
- We need to find the true mass \( m_0 \) of the body.

2. **Buoyant Force Calculation**:
- When the body is submerged in air, it experiences a buoyant force. The buoyant force \( F_b \) can be calculated using Archimedes' principle:
\[ F_b = \text{density of air} \times \text{volume of the body} \times g \]
- The volume \( V \) of the body can be expressed as:
\[ V = \frac{m_0}{d} \]
- Therefore, the buoyant force becomes:
\[ F_b = d \times \frac{m_0}{d} \times g = m_0 g \]

3. **Apparent Weight Calculation**:
- The apparent weight \( W_a \) of the body in air is given by:
\[ W_a = \text{true weight} - \text{buoyant force} = m_0 g - F_b \]
- Substituting the buoyant force:
\[ W_a = m_0 g - m_0 g = 0 \]
- This indicates that the body is in equilibrium with the counterpoised mass.

4. **Equating Apparent Weights**:
- The apparent weight of the body must equal the apparent weight of the counterpoised mass \( m \) of density \( d_1 \):
\[ m g - F_b' = 0 \]
- The buoyant force on the counterpoised mass can be calculated similarly:
\[ F_b' = d \times \frac{m}{d_1} \times g \]
- Thus, we have:
\[ m g - d \times \frac{m}{d_1} \times g = 0 \]

5. **Setting Up the Equation**:
- Since both apparent weights are equal, we can write:
\[ m_0 g - d \times \frac{m_0}{d} \times g = m g - d \times \frac{m}{d_1} \times g \]
- Canceling \( g \) from both sides:
\[ m_0 - \frac{m_0 d}{d} = m - \frac{m d}{d_1} \]

6. **Solving for True Mass**:
- Rearranging gives:
\[ m_0 \left(1 - \frac{d}{d_1}\right) = m \left(1 - \frac{d}{d_2}\right) \]
- Therefore, the true mass \( m_0 \) can be expressed as:
\[ m_0 = m \frac{(1 - \frac{d}{d_2})}{(1 - \frac{d}{d_1})} \]

### Final Answer:
The true mass of the body is:
\[ m_0 = m \frac{(1 - \frac{d}{d_2})}{(1 - \frac{d}{d_1})} \]