A vessel contains oil (density `=0.8gm//cm^3`) over mercury (density `=13.6gm cm^3`). A homogeneous sphere floats with half its volume immersed in mercury and the other half in oil. The density of the material of the sphere in `gm//cm^3` is

Options

A. `3.3`
B. `6.4`
C. `7.2`
D. `12.8`

Explanation

To solve the problem, we need to find the density of the homogeneous sphere that is floating in a vessel containing oil and mercury. The sphere is half-immersed in mercury and half-immersed in oil.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a sphere floating in two fluids: oil (density = 0.8 g/cm³) and mercury (density = 13.6 g/cm³).
- The sphere is half-immersed in mercury and half-immersed in oil.

2. **Identify the Forces Acting on the Sphere**:
- The weight of the sphere (W) acts downwards.
- The buoyant force (B) acts upwards and is equal to the weight of the fluid displaced by the submerged part of the sphere.

3. **Calculate the Buoyant Force**:
- The volume of the sphere is V, so the volume submerged in mercury is V/2 and the volume submerged in oil is also V/2.
- The buoyant force from mercury (B_Hg) is given by:
\[ B_{Hg} = \text{density of mercury} \times \text{volume submerged in mercury} \times g = 13.6 \times \frac{V}{2} \times g \]
- The buoyant force from oil (B_oil) is given by:
\[ B_{oil} = \text{density of oil} \times \text{volume submerged in oil} \times g = 0.8 \times \frac{V}{2} \times g \]

4. **Total Buoyant Force**:
- The total buoyant force (B) acting on the sphere is the sum of the buoyant forces from both fluids:
\[ B = B_{Hg} + B_{oil} = \left(13.6 \times \frac{V}{2} \times g\right) + \left(0.8 \times \frac{V}{2} \times g\right) \]
- Simplifying this gives:
\[ B = \left(13.6 + 0.8\right) \times \frac{V}{2} \times g = 14.4 \times \frac{V}{2} \times g \]

5. **Weight of the Sphere**:
- The weight of the sphere (W) is given by:
\[ W = \text{density of sphere} \times \text{volume of sphere} \times g = \rho_s \times V \times g \]

6. **Equating Weight and Buoyant Force**:
- For the sphere to float, the weight of the sphere must equal the total buoyant force:
\[ \rho_s \times V \times g = 14.4 \times \frac{V}{2} \times g \]

7. **Canceling Common Terms**:
- We can cancel V and g from both sides (assuming V is not zero and g is constant):
\[ \rho_s = 14.4 \times \frac{1}{2} = 7.2 \text{ g/cm}^3 \]

8. **Final Result**:
- The density of the material of the sphere is **7.2 g/cm³**.