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. (a) 3.3
B. (b) 6.4
C. (c) 7.2
D. (d) 12.8

Explanation

To find the density of the material of the sphere, we can use the principle of buoyancy. The sphere is floating with half of its volume immersed in mercury and half in oil, which allows us to set up an equation based on the forces acting on the sphere.

### Step-by-Step Solution:

1. **Identify the densities**:
- Density of oil, \( \rho_{\text{oil}} = 0.8 \, \text{g/cm}^3 \)
- Density of mercury, \( \rho_{\text{Hg}} = 13.6 \, \text{g/cm}^3 \)

2. **Let the volume of the sphere be \( V \)**. Since half of the sphere is immersed in mercury and half in oil, the volume immersed in each fluid is:
- Volume in mercury, \( V_{\text{Hg}} = \frac{V}{2} \)
- Volume in oil, \( V_{\text{oil}} = \frac{V}{2} \)

3. **Calculate the weight of the sphere**:
- The weight of the sphere, \( W = \text{density of sphere} \times V \times g \)
- Let the density of the sphere be \( \rho_s \). Therefore, \( W = \rho_s \cdot V \cdot g \)

4. **Calculate the buoyant force**:
- The buoyant force from the mercury is given by Archimedes' principle:
\[ F_{\text{buoyant, Hg}} = \rho_{\text{Hg}} \cdot V_{\text{Hg}} \cdot g = \rho_{\text{Hg}} \cdot \left(\frac{V}{2}\right) \cdot g = \frac{13.6 \cdot V \cdot g}{2} \]
- The buoyant force from the oil is:
\[ F_{\text{buoyant, oil}} = \rho_{\text{oil}} \cdot V_{\text{oil}} \cdot g = \rho_{\text{oil}} \cdot \left(\frac{V}{2}\right) \cdot g = \frac{0.8 \cdot V \cdot g}{2} \]

5. **Total buoyant force**:
- The total buoyant force acting on the sphere is the sum of the buoyant forces from both fluids:
\[ F_{\text{buoyant}} = F_{\text{buoyant, Hg}} + F_{\text{buoyant, oil}} = \frac{13.6 \cdot V \cdot g}{2} + \frac{0.8 \cdot V \cdot g}{2} \]
- Simplifying this gives:
\[ F_{\text{buoyant}} = \frac{(13.6 + 0.8) \cdot V \cdot g}{2} = \frac{14.4 \cdot V \cdot g}{2} \]

6. **Set the weight equal to the buoyant force**:
- Since the sphere is floating, the weight of the sphere is equal to the total buoyant force:
\[ \rho_s \cdot V \cdot g = \frac{14.4 \cdot V \cdot g}{2} \]

7. **Cancel \( V \cdot g \) from both sides** (assuming \( V \) and \( g \) are not zero):
\[ \rho_s = \frac{14.4}{2} = 7.2 \, \text{g/cm}^3 \]

### Final Answer:
The density of the material of the sphere is \( \rho_s = 7.2 \, \text{g/cm}^3 \).