Express the relationship between the C.G.S. and S.I. units of density.

Answer

To express the relationship between the C.G.S. (Centimeter-Gram-Second) and S.I. (International System of Units) units of density, we can follow these steps:

### Step 1: Understand the formula for density
Density (ρ) is defined as mass (M) divided by volume (V):
\[ \rho = \frac{M}{V} \]

### Step 2: Identify the units in S.I.
In the S.I. system:
- The unit of mass is kilograms (kg).
- The unit of length is meters (m).
- Therefore, the unit of volume is cubic meters (m³).

Thus, the S.I. unit of density is:
\[ \text{Density (S.I.)} = \frac{\text{kg}}{\text{m}^3} \]

### Step 3: Identify the units in C.G.S.
In the C.G.S. system:
- The unit of mass is grams (g).
- The unit of length is centimeters (cm).
- Therefore, the unit of volume is cubic centimeters (cm³).

Thus, the C.G.S. unit of density is:
\[ \text{Density (C.G.S.)} = \frac{\text{g}}{\text{cm}^3} \]

### Step 4: Convert S.I. units to C.G.S. units
To convert from S.I. to C.G.S.:
1 kg = 1000 g
1 m = 100 cm
Therefore, 1 m³ = (100 cm)³ = 1,000,000 cm³ = 10^6 cm³.

Now, substituting these values into the S.I. unit of density:
\[ 1 \, \text{kg/m}^3 = \frac{1000 \, \text{g}}{10^6 \, \text{cm}^3} = \frac{1000}{10^6} \, \text{g/cm}^3 = \frac{1}{1000} \, \text{g/cm}^3 \]

### Step 5: Establish the final relationship
From the above calculation, we can see that:
\[ 1 \, \text{kg/m}^3 = \frac{1}{1000} \, \text{g/cm}^3 \]
This means that:
\[ 1000 \, \text{kg/m}^3 = 1 \, \text{g/cm}^3 \]

### Conclusion
The relationship between the C.G.S. and S.I. units of density is:
\[ 1 \, \text{g/cm}^3 = 1000 \, \text{kg/m}^3 \]