A shell of relative density `27/9` w.r.t water is just submerged in water. If its inner & outer radius is r and R then r/R is

Options

A. `(1/3)^(1/3)`
B. `(2/3)^(1/3)`
C. `(3/4)^(1/3)`
D. `(5/9)^(1/3)`

Answer

To solve the problem, we need to find the ratio of the inner radius (r) to the outer radius (R) of a shell that is just submerged in water. The shell has a relative density of \( \frac{27}{9} \) with respect to water.

### Step-by-Step Solution:

1. **Understanding Relative Density**:
The relative density (RD) of the shell is given as:
\[ RD = \frac{\text{Density of shell} (\rho_s)}{\text{Density of water} (\rho_w)} = \frac{27}{9} = 3 \]
This means:
\[ \rho_s = 3 \rho_w \]

2. **Buoyant Force and Weight**:
Since the shell is just submerged, the buoyant force equals the weight of the shell:
\[ \text{Buoyant Force} = \text{Weight of the shell} \]

3. **Calculating Buoyant Force**:
The buoyant force can be expressed as:
\[ \text{Buoyant Force} = \text{Volume of displaced water} \times \rho_w \times g \]
The volume of displaced water is equal to the volume of the outer shell:
\[ V_{\text{displaced}} = \frac{4}{3} \pi R^3 \]
Thus,
\[ \text{Buoyant Force} = \frac{4}{3} \pi R^3 \rho_w g \]

4. **Calculating Weight of the Shell**:
The weight of the shell can be expressed as:
\[ \text{Weight of the shell} = \text{Volume of the shell} \times \rho_s \times g \]
The volume of the shell is:
\[ V_{\text{shell}} = \frac{4}{3} \pi (R^3 - r^3) \]
Thus,
\[ \text{Weight of the shell} = \frac{4}{3} \pi (R^3 - r^3) \rho_s g \]

5. **Setting Buoyant Force Equal to Weight**:
Setting the two expressions equal gives:
\[ \frac{4}{3} \pi R^3 \rho_w g = \frac{4}{3} \pi (R^3 - r^3) \rho_s g \]
We can cancel \( \frac{4}{3} \pi g \) from both sides:
\[ R^3 \rho_w = (R^3 - r^3) \rho_s \]

6. **Substituting Density**:
Substitute \( \rho_s = 3 \rho_w \):
\[ R^3 \rho_w = (R^3 - r^3)(3 \rho_w) \]
Dividing both sides by \( \rho_w \):
\[ R^3 = 3(R^3 - r^3) \]

7. **Simplifying the Equation**:
Expanding and rearranging gives:
\[ R^3 = 3R^3 - 3r^3 \]
\[ 3r^3 = 2R^3 \]
\[ \frac{r^3}{R^3} = \frac{2}{3} \]

8. **Finding the Ratio**:
Taking the cube root on both sides:
\[ \frac{r}{R} = \left(\frac{2}{3}\right)^{\frac{1}{3}} \]

### Final Result:
Thus, the ratio of the inner radius to the outer radius is:
\[ \frac{r}{R} = \left(\frac{2}{3}\right)^{\frac{1}{3}} \]