The value fo universal gravitationla constant `G` in `CGS` system is `6.67xx10^(-8)` dyne `cm^(2) g^(-2)`. Its value in `SI` system is

Options

A. `6.67xx10^(-11) Nm^(2) kg^(-2)`
B. `6.67xx10^(-5) Nm^(2) kg^(-2)`
C. `6.67xx10^(-10) Nm^(2) kg^(-2)`
D. `6.67xx10^(-9) Nm^(2) kg^(-2)`

Explanation

To convert the universal gravitational constant \( G \) from the CGS system to the SI system, we will follow these steps:

### Step 1: Write down the given value of \( G \) in CGS units
The value of the universal gravitational constant \( G \) in the CGS system is given as:
\[ G = 6.67 \times 10^{-8} \, \text{dyne} \, \text{cm}^2 \, \text{g}^{-2} \]

### Step 2: Convert the units from CGS to SI
We need to convert the units of dyne, centimeter, and gram to their respective SI units.

1. **Convert dyne to Newton**:
\[ 1 \, \text{dyne} = 10^{-5} \, \text{N} \]

2. **Convert centimeter to meter**:
\[ 1 \, \text{cm} = 10^{-2} \, \text{m} \]

3. **Convert gram to kilogram**:
\[ 1 \, \text{g} = 10^{-3} \, \text{kg} \]

### Step 3: Substitute the conversions into the expression for \( G \)
Now we will substitute these conversions into the expression for \( G \):
\[ G = 6.67 \times 10^{-8} \, \text{dyne} \, \text{cm}^2 \, \text{g}^{-2} \]
Substituting the conversions:
\[ G = 6.67 \times 10^{-8} \left(10^{-5} \, \text{N}\right) \left(10^{-2} \, \text{m}\right)^2 \left(10^{-3} \, \text{kg}\right)^{-2} \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ G = 6.67 \times 10^{-8} \times 10^{-5} \times (10^{-2})^2 \times (10^{-3})^{-2} \]
Calculating the powers:
\[ (10^{-2})^2 = 10^{-4} \]
\[ (10^{-3})^{-2} = 10^{6} \]
Now substituting these values back:
\[ G = 6.67 \times 10^{-8} \times 10^{-5} \times 10^{-4} \times 10^{6} \]

### Step 5: Combine the powers of ten
Now we combine the powers of ten:
\[ G = 6.67 \times 10^{-8 - 5 - 4 + 6} \]
Calculating the exponent:
\[ -8 - 5 - 4 + 6 = -11 \]
Thus:
\[ G = 6.67 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2} \]

### Final Answer
The value of the universal gravitational constant \( G \) in the SI system is:
\[ G = 6.67 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2} \]

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