A hollow sphere of volume V is floating on water surface with half immersed in it. What should be the minimum volume of water poured inside the sphere so that the sphere now sinks into the water ?

Options

A. V/2
B. V/3
C. V/4
D. V

Answer

To solve the problem of determining the minimum volume of water that needs to be poured inside a hollow sphere so that it sinks completely in water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
The hollow sphere has a volume \( V \) and is floating on water with half of its volume immersed. This means that the volume of water displaced by the sphere is \( \frac{V}{2} \).

2. **Apply Archimedes' Principle**:
According to Archimedes' principle, the buoyant force acting on the sphere is equal to the weight of the water displaced. The weight of the water displaced can be expressed as:
\[ F_b = \text{Volume of water displaced} \times \text{Density of water} \times g = \frac{V}{2} \cdot \rho \cdot g \]
where \( \rho \) is the density of water and \( g \) is the acceleration due to gravity.

3. **Calculate the Weight of the Sphere**:
The weight of the sphere can be expressed as:
\[ W_s = m \cdot g \]
where \( m \) is the mass of the sphere.

4. **Determine the Weight After Adding Water**:
Let \( V_0 \) be the volume of water added to the sphere. The weight of the sphere after adding water becomes:
\[ W_s' = m \cdot g + V_0 \cdot \rho \cdot g \]

5. **Condition for Sinking**:
For the sphere to sink completely, the weight of the sphere plus the weight of the water inside must equal the buoyant force when the sphere is fully submerged. The buoyant force when fully submerged is:
\[ F_b' = V \cdot \rho \cdot g \]

6. **Set Up the Equation**:
We set the total weight equal to the buoyant force:
\[ m \cdot g + V_0 \cdot \rho \cdot g = V \cdot \rho \cdot g \]

7. **Simplify the Equation**:
Dividing through by \( g \) (assuming \( g \neq 0 \)):
\[ m + V_0 \cdot \rho = V \cdot \rho \]

8. **Rearranging for \( V_0 \)**:
Rearranging gives:
\[ V_0 \cdot \rho = V \cdot \rho - m \]
Thus,
\[ V_0 = \frac{V \cdot \rho - m}{\rho} \]

9. **Using the Initial Condition**:
Since the sphere is floating half immersed, we know that:
\[ m = \frac{V}{2} \cdot \rho \]
Substituting this into the equation for \( V_0 \):
\[ V_0 = \frac{V \cdot \rho - \frac{V}{2} \cdot \rho}{\rho} = \frac{V \cdot \rho - \frac{V}{2} \cdot \rho}{\rho} = \frac{V}{2} \]

10. **Conclusion**:
Therefore, the minimum volume of water that needs to be poured inside the sphere for it to sink completely is:
\[ V_0 = \frac{V}{2} \]

### Final Answer:
The minimum volume of water that needs to be poured inside the sphere is \( \frac{V}{2} \).