A ball is made of a material of density `rho` where `rho_(oil)ltrholtrho_(water)` with `rho_(oil)` and `rho_(water)` representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the behavior of a ball made of a material with a density \(\rho\) in a mixture of oil and water, where the density of oil is less than that of water.

### Step-by-step Solution:

1. **Understanding the Densities**:
- We are given that the density of the ball (\(\rho\)) is greater than the density of oil (\(\rho_{oil}\)) but less than the density of water (\(\rho_{water}\)).
- This can be summarized as:
\[ \rho_{oil} < \rho < \rho_{water} \]

2. **Behavior in Oil**:
- Since the density of the ball is greater than the density of oil, the ball will sink in the oil. This is because objects with greater density than the fluid they are in will sink.

3. **Behavior in Water**:
- Since the density of the ball is less than the density of water, the ball will float in the water. Objects with lesser density than the fluid they are in will float.

4. **Arrangement of Fluids**:
- In a container with immiscible fluids (oil and water), the less dense fluid (oil) will float on top of the denser fluid (water). Therefore, the arrangement will be:
- Oil on top
- Water below

5. **Equilibrium Position of the Ball**:
- In this scenario, the ball will sink to the bottom of the oil layer and float at the interface between the oil and water. Therefore, the ball will be positioned in such a way that it is partially submerged in the oil and partially submerged in the water.

6. **Identifying the Correct Diagram**:
- Based on the above analysis, we need to find a diagram that shows:
- Oil on top
- Water below
- The ball sinking in the oil but floating in the water.
- The correct option will depict the ball floating at the interface of the oil and water.

### Conclusion:
The correct answer is option 2, which shows the ball sinking in the oil and floating in the water.

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