If`p_A and P_B` are the densities of the material of body and the liquid respectively then the body can float only when

Options

A. `p_A=p_B`
B. `p_Altp_B`
C. `p_Agtp_B`
D. `p_A=2P_B`

Answer

To determine the condition under which a body can float in a liquid, we can analyze the relationship between the densities of the body and the liquid. Here’s a step-by-step solution to the problem:

### Step-by-Step Solution:

1. **Understanding the Concept of Floatation**:
- A body will float in a liquid if the buoyant force acting on it is equal to its weight. The buoyant force is determined by the density of the liquid and the volume of the liquid displaced by the body.

2. **Defining Densities**:
- Let \( P_A \) be the density of the body (solid).
- Let \( P_B \) be the density of the liquid.

3. **Applying Archimedes' Principle**:
- According to Archimedes' principle, the buoyant force \( F_b \) acting on the body is equal to the weight of the liquid displaced by the body. This can be expressed as:
\[ F_b = P_B \cdot V \cdot g \]
- Where \( V \) is the volume of the body submerged in the liquid and \( g \) is the acceleration due to gravity.

4. **Setting Up the Condition for Floatation**:
- For the body to float, the buoyant force must be equal to the weight of the body:
\[ F_b = P_A \cdot V \cdot g \]
- Therefore, we have:
\[ P_B \cdot V \cdot g = P_A \cdot V \cdot g \]

5. **Simplifying the Equation**:
- Since \( g \) and \( V \) are common on both sides, we can simplify the equation to:
\[ P_B = P_A \]
- This indicates that if the density of the liquid is equal to the density of the body, the body will float.

6. **Considering Other Conditions**:
- However, if the density of the liquid is greater than the density of the body, the body will also float. Thus, we can conclude that:
\[ P_B \geq P_A \]

### Conclusion:
A body can float in a liquid if the density of the liquid \( P_B \) is greater than or equal to the density of the body \( P_A \). Therefore, the condition for floatation is:
\[ P_B \geq P_A \]