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

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, given that its relative density with respect to water is \( \frac{27}{9} \).

### Step-by-Step Solution:

1. **Understanding Relative Density**:
The relative density (specific gravity) is defined as the ratio of the density of a substance to the density of a reference substance (in this case, water).
\[ \text{Relative Density} = \frac{\text{Density of Shell}}{\text{Density of Water}} = \frac{27}{9} = 3 \]
Therefore, the density of the shell (\( \rho_s \)) can be expressed as:
\[ \rho_s = 3 \cdot \rho_w \]
where \( \rho_w \) is the density of water.

2. **Volume of the Shell**:
The volume of the shell can be calculated using the formula for the volume of a hollow sphere:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]

3. **Mass of the Shell**:
The mass of the shell can be calculated using the density:
\[ m = \rho_s \cdot V = \rho_s \cdot \frac{4}{3} \pi (R^3 - r^3) \]
Substituting \( \rho_s \):
\[ m = 3 \cdot \rho_w \cdot \frac{4}{3} \pi (R^3 - r^3) = 4 \pi \rho_w (R^3 - r^3) \]

4. **Buoyant Force**:
According to Archimedes' principle, the buoyant force (\( F_b \)) acting on the shell when it is submerged is equal to the weight of the water displaced by the shell:
\[ F_b = \rho_w \cdot V_{displaced} \cdot g = \rho_w \cdot \frac{4}{3} \pi R^3 \cdot g \]

5. **Equating Weight and Buoyant Force**:
For the shell to be just submerged, the weight of the shell must equal the buoyant force:
\[ 4 \pi \rho_w (R^3 - r^3) \cdot g = \rho_w \cdot \frac{4}{3} \pi R^3 \cdot g \]
Canceling \( 4 \pi g \) from both sides:
\[ \rho_w (R^3 - r^3) = \frac{1}{3} \rho_w R^3 \]

6. **Simplifying the Equation**:
Dividing both sides by \( \rho_w \):
\[ R^3 - r^3 = \frac{1}{3} R^3 \]
Rearranging gives:
\[ R^3 - \frac{1}{3} R^3 = r^3 \]
\[ \frac{2}{3} R^3 = r^3 \]

7. **Finding the Ratio**:
Taking the cube root of both sides:
\[ r = R \left( \frac{2}{3} \right)^{1/3} \]
Therefore, the ratio \( \frac{r}{R} \) is:
\[ \frac{r}{R} = \left( \frac{2}{3} \right)^{1/3} \]

### Final Answer:
\[ \frac{r}{R} = \left( \frac{2}{3} \right)^{1/3} \]