Two different object are completely immersed in water and undergo same loss in weight. Is it necessary that the weights of these objects in air be also the same?

Options

A. A and B have same weight in air
B. A and B have equal volumes
C. The densities of the materials of A and B are the same
D. A and B are immersed to the same depth inside water .

Answer

To determine whether two different objects that undergo the same loss in weight when immersed in water must also have the same weight in air, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding Loss of Weight in Water**:
When an object is immersed in water, it experiences a loss of weight due to the buoyant force acting on it. The buoyant force is equal to the weight of the water displaced by the object.

2. **Buoyant Force Calculation**:
The buoyant force (B) can be expressed as:
\[ B = \text{Weight of water displaced} = \rho_{water} \cdot g \cdot V \]
where \( \rho_{water} \) is the density of water, \( g \) is the acceleration due to gravity, and \( V \) is the volume of the object.

3. **Loss of Weight**:
The loss of weight (L) of the object when immersed in water is given by:
\[ L = B = \rho_{water} \cdot g \cdot V \]
This shows that the loss of weight depends only on the volume of the object and the density of the fluid (water in this case).

4. **Different Densities**:
Let’s consider two objects with densities \( D_1 \) and \( D_2 \). If both objects have the same volume \( V \) and undergo the same loss of weight \( L \), then:
\[ L = \rho_{water} \cdot g \cdot V \]
This means both objects displace the same volume of water, resulting in the same buoyant force.

5. **Weight in Air**:
The weight of each object in air can be expressed as:
\[ W_1 = D_1 \cdot V \cdot g \quad \text{and} \quad W_2 = D_2 \cdot V \cdot g \]
Here, \( W_1 \) and \( W_2 \) are the weights of the objects in air.

6. **Comparing Weights**:
Since the densities \( D_1 \) and \( D_2 \) can be different, it follows that:
\[ W_1 = D_1 \cdot V \cdot g \quad \text{and} \quad W_2 = D_2 \cdot V \cdot g \]
If \( D_1 \neq D_2 \), then \( W_1 \neq W_2 \). Thus, the weights of the objects in air do not have to be the same even if they experience the same loss of weight in water.

### Conclusion:
It is not necessary that the weights of the two objects in air be the same, even if they undergo the same loss in weight when immersed in water.