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 \( G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \) into the units of \( g^{-1} \, \text{cm}^3 \, \text{s}^{-2} \), we will follow these steps:

### Step 1: Understand the Units
The given unit of \( G \) is in terms of Newtons (N), meters (m), and kilograms (kg). We need to convert these units into the CGS (centimeter-gram-second) system.

### Step 2: Convert Newton to CGS
Recall that:
- \( 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \)

In CGS:
- \( 1 \, \text{kg} = 1000 \, \text{g} \)
- \( 1 \, \text{m} = 100 \, \text{cm} \)
- \( 1 \, \text{s} = 1 \, \text{s} \)

Thus, we can convert \( 1 \, \text{N} \):
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 = 1000 \, \text{g} \cdot (100 \, \text{cm}/\text{s}^2) = 100000 \, \text{g} \cdot \text{cm/s}^2 = 10^5 \, \text{g} \cdot \text{cm/s}^2 \]

### Step 3: Substitute into G
Now substituting this back into the expression for \( G \):
\[ G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 = 6.67 \times 10^{-11} \cdot (10^5 \, \text{g} \cdot \text{cm/s}^2) \cdot (100^2 \, \text{cm}^2/(1000^2 \, \text{g}^2)) \]

### Step 4: Simplify the Expression
Now we simplify:
\[ G = 6.67 \times 10^{-11} \cdot (10^5) \cdot (10^4) \, \text{g}^{-1} \cdot \text{cm}^3 \cdot \text{s}^{-2} \]
\[ = 6.67 \times 10^{-11} \cdot 10^9 \, \text{g}^{-1} \cdot \text{cm}^3 \cdot \text{s}^{-2} \]

### Step 5: Calculate the Final Value
Calculating the numerical value:
\[ 6.67 \times 10^{-11} \cdot 10^9 = 6.67 \times 10^{-2} \]

### Final Result
Thus, the value of \( G \) in units of \( g^{-1} \, \text{cm}^3 \, \text{s}^{-2} \) is:
\[ G = 6.67 \times 10^{-2} \, g^{-1} \, \text{cm}^3 \, \text{s}^{-2} \]

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