The dimensions of universal gravitational constant are ____

Options

A. `[M^(-1)L^(3)T^(-2)]`
B. `[ML^(2)T^(-1)]`
C. `[M^(-2)L^(3)T^(-2)]`
D. `[M^(-2)L^(2)T^(-1)]`

Explanation

To find the dimensions of the universal gravitational constant \( G \), we start with the formula for gravitational force:

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

Where:
- \( F \) is the gravitational force,
- \( G \) is the universal gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( d \) is the distance between the centers of the two masses.

### Step 1: Rearranging the formula
We can rearrange the formula to express \( G \):

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

### Step 2: Identifying the dimensions
Now, we need to identify the dimensions of each variable in the equation:

1. **Force \( F \)** has the dimensions:
\[ [F] = M L T^{-2} \]

2. **Distance \( d \)** has the dimensions:
\[ [d] = L \]

3. **Mass \( m_1 \) and \( m_2 \)** have the dimensions:
\[ [m_1] = [m_2] = M \]

### Step 3: Substituting dimensions into the equation
Now, substituting the dimensions into the rearranged formula for \( G \):

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

### Step 4: Simplifying the expression
Now, simplify the expression:

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

This simplifies to:

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

### Conclusion
Thus, the dimensions of the universal gravitational constant \( G \) are:

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

---