To find the ratio of the SI unit of the gravitational constant \( G \) to its CGS unit, we will follow these steps:
### Step 1: Identify the SI unit of \( G \)
The SI unit of the gravitational constant \( G \) is given as:
\[
G = 6.67 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}
\]
### Step 2: Convert the SI unit to CGS unit
To convert the SI unit to CGS unit, we need to convert meters to centimeters and kilograms to grams.
1. **Convert meters to centimeters:**
\[
1 \, \text{m} = 100 \, \text{cm} \quad \Rightarrow \quad 1 \, \text{m}^3 = (100 \, \text{cm})^3 = 100^3 \, \text{cm}^3 = 10^6 \, \text{cm}^3
\]
2. **Convert kilograms to grams:**
\[
1 \, \text{kg} = 1000 \, \text{g} \quad \Rightarrow \quad 1 \, \text{kg}^{-1} = \frac{1}{1000} \, \text{g}^{-1} = 10^{-3} \, \text{g}^{-1}
\]
3. **The time unit remains the same:**
\[
1 \, \text{s} = 1 \, \text{s}
\]
Putting it all together, we convert \( G \) to CGS units:
\[
G = 6.67 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} = 6.67 \times 10^{-11} \times 10^6 \, \text{cm}^3 \times 10^{-3} \, \text{g}^{-1} \, \text{s}^{-2}
\]
\[
= 6.67 \times 10^{-5} \, \text{cm}^3 \, \text{g}^{-1} \, \text{s}^{-2}
\]
### Step 3: Write the CGS unit of \( G \)
The CGS unit of \( G \) is:
\[
G = 6.67 \times 10^{-8} \, \text{cm}^3 \, \text{g}^{-1} \, \text{s}^{-2}
\]
### Step 4: Calculate the ratio of SI unit to CGS unit
Now, we can find the ratio of the SI unit to the CGS unit:
\[
\text{Ratio} = \frac{6.67 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}}{6.67 \times 10^{-8} \, \text{cm}^3 \, \text{g}^{-1} \, \text{s}^{-2}}
\]
### Step 5: Simplify the ratio
The \( 6.67 \) and \( \text{s}^{-2} \) cancel out:
\[
\text{Ratio} = \frac{10^{-11} \, \text{m}^3 \, \text{kg}^{-1}}{10^{-8} \, \text{cm}^3 \, \text{g}^{-1}}
\]
Now, converting the units:
\[
\text{m} = 100 \, \text{cm} \quad \Rightarrow \quad \text{m}^3 = 10^6 \, \text{cm}^3
\]
\[
\text{kg} = 1000 \, \text{g} \quad \Rightarrow \quad \text{kg}^{-1} = 10^{-3} \, \text{g}^{-1}
\]
Substituting these conversions into the ratio:
\[
\text{Ratio} = \frac{10^{-11} \times 10^6 \, \text{cm}^3 \times 10^{-3} \, \text{g}^{-1}}{10^{-8} \, \text{cm}^3 \, \text{g}^{-1}}
\]
\[
= \frac{10^{-11 + 6 - 3}}{10^{-8}} = \frac{10^{-8}}{10^{-8}} = 1
\]
### Final Answer
The ratio of the SI unit of \( G \) to the CGS unit of \( G \) is:
\[
\text{Ratio} = 1
\]
