A suitable unit for gravitational constant is

Options

A. `kg-msec^(-1)`
B. `Nm^(-1)sec`
C. `Nm^2kg^(-2)`
D. `kgmsec^(-1)`

Explanation

To find a suitable unit for the gravitational constant (G), we start from the formula that relates gravitational force (F) to G:

\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

### Step 1: Rearranging the Formula
We can rearrange this formula to express G in terms of F, m1, m2, and r:

\[ G = \frac{F \cdot r^2}{m_1 \cdot m_2} \]

### Step 2: Identifying the Dimensions
Next, we need to identify the dimensions of each term in the equation:

- The dimension of force (F) is given by Newton (N). In terms of basic units, 1 Newton is defined as \( 1 \, \text{kg} \cdot \text{m/s}^2 \). Thus, the dimension of force is:
\[ [F] = [M][L][T^{-2}] \]

- The distance (r) has the dimension of length, which is:
\[ [r] = [L] \]

- The masses (m1 and m2) have the dimension of mass:
\[ [m_1] = [M], \quad [m_2] = [M] \]

### Step 3: Substituting the Dimensions
Now we substitute these dimensions back into the equation for G:

\[ [G] = \frac{[F] \cdot [r]^2}{[m_1] \cdot [m_2]} = \frac{[M][L][T^{-2}] \cdot [L]^2}{[M] \cdot [M]} \]

### Step 4: Simplifying the Dimensions
Now simplifying the expression:

\[ [G] = \frac{[M][L^3][T^{-2}]}{[M^2]} = [M^{-1}][L^3][T^{-2}] \]

### Step 5: Converting to Suitable Units
In terms of SI units, we can express the dimensions of G as:

\[ [G] = \frac{\text{m}^3}{\text{kg} \cdot \text{s}^2} \]

Thus, the suitable unit for the gravitational constant G is:

\[ \text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2} \]

### Conclusion
After analyzing the options provided, we can identify the correct unit for the gravitational constant based on our derived expression.

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