The value of universal gravitational constant `G=6.67xx10^(-11)Nm^(2)kg^(-2)`. The value of G in units of `g^(-1)cm^(3)s^(-2)` is

Options

A. `6.67xx10^(-8)`
B. `6.67xx10^(-7)`
C. `6.67xx10^(-9)`
D. `6.67xx10^(-10)`

Explanation

To convert the universal gravitational constant \( G \) from its standard SI units to the units of \( g^{-1} \, cm^3 \, s^{-2} \), we will follow these steps:

### Step 1: Write down the given value of \( G \)
The value of the universal gravitational constant is given as:
\[ G = 6.67 \times 10^{-11} \, \text{N} \, m^2 \, kg^{-2} \]

### Step 2: Convert Newtons to base units
Recall that 1 Newton (N) can be expressed in terms of base SI units:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \]
Thus, we can rewrite \( G \):
\[ G = 6.67 \times 10^{-11} \, \text{kg} \cdot \text{m/s}^2 \cdot m^2 \cdot kg^{-2} = 6.67 \times 10^{-11} \, \text{kg}^{-1} \cdot m^2 \cdot s^{-2} \]

### Step 3: Substitute the units of mass and length
Next, we need to convert kilograms to grams and meters to centimeters:
- \( 1 \, \text{kg} = 1000 \, \text{g} \)
- \( 1 \, \text{m} = 100 \, \text{cm} \)

Substituting these into our expression for \( G \):
\[ G = 6.67 \times 10^{-11} \cdot \left( \frac{1}{1000 \, \text{g}} \right) \cdot \left( 100 \, \text{cm} \right)^2 \cdot s^{-2} \]

### Step 4: Simplify the expression
Now, let's simplify the expression:
\[ G = 6.67 \times 10^{-11} \cdot \frac{10000 \, \text{cm}^2}{1000 \, \text{g}} \cdot s^{-2} \]
\[ G = 6.67 \times 10^{-11} \cdot \frac{10000}{1000} \cdot \frac{1}{\text{g}} \cdot \text{cm}^2 \cdot s^{-2} \]
\[ G = 6.67 \times 10^{-11} \cdot 10 \cdot \frac{1}{\text{g}} \cdot \text{cm}^2 \cdot s^{-2} \]
\[ G = 6.67 \times 10^{-10} \cdot \frac{1}{\text{g}} \cdot \text{cm}^2 \cdot s^{-2} \]

### Step 5: Final conversion to required units
Now we need to express \( G \) in the required units of \( g^{-1} \, cm^3 \, s^{-2} \):
\[ G = 6.67 \times 10^{-10} \cdot \frac{1}{\text{g}} \cdot \text{cm}^2 \cdot s^{-2} \]
To convert \( \text{cm}^2 \) to \( \text{cm}^3 \), we can multiply by \( \text{cm} \):
\[ G = 6.67 \times 10^{-10} \cdot \frac{1}{\text{g}} \cdot \text{cm}^3 \cdot s^{-2} \cdot \frac{1}{\text{cm}} \]
Thus, we have:
\[ G = 6.67 \times 10^{-8} \cdot g^{-1} \cdot \text{cm}^3 \cdot s^{-2} \]

### Conclusion
The value of the universal gravitational constant \( G \) in units of \( g^{-1} \, cm^3 \, s^{-2} \) is:
\[ G = 6.67 \times 10^{-8} \, g^{-1} \, cm^3 \, s^{-2} \]