The quantity having negative dimensions in mass is

Options

A. Gravitational Potential
B. Gravitational constant
C. Acceleration due to gravity
D. None of the above

Answer

To solve the question of which quantity has negative dimensions in mass, we will analyze each option provided.

### Step-by-Step Solution:

1. **Understanding the Question**:
We need to identify which of the given quantities has negative dimensions in mass. The quantities given are gravitational potential, gravitational constant, acceleration due to gravity, and none of the above.

2. **Analyzing Gravitational Potential**:
- Gravitational potential (V) is given by the formula:
\[ V = \frac{GM}{r} \]
- Here, \(G\) is the gravitational constant, \(M\) is mass, and \(r\) is distance.
- The dimensions of gravitational potential can be derived as follows:
\[ [V] = \frac{[G][M]}{[L]} = \frac{[M^{-1}L^3T^{-2}][M]}{[L]} = [M^0L^2T^{-2}] \]
- Thus, the dimensions of gravitational potential are \(M^0 L^2 T^{-2}\) (no negative mass dimension).

3. **Analyzing Gravitational Constant**:
- The gravitational constant \(G\) is defined in the context of Newton's law of gravitation:
\[ F = \frac{GM_1M_2}{r^2} \]
- Rearranging gives:
\[ G = \frac{Fr^2}{M_1M_2} \]
- The dimensions of \(G\) can be derived as follows:
\[ [G] = \frac{[F][L^2]}{[M^2]} = \frac{[M L T^{-2}][L^2]}{[M^2]} = [M^{-1}L^3T^{-2}] \]
- Thus, the dimensions of the gravitational constant are \(M^{-1} L^3 T^{-2}\) (which includes a negative mass dimension).

4. **Analyzing Acceleration Due to Gravity**:
- The acceleration due to gravity \(g\) is given by:
\[ g = \frac{GM}{r^2} \]
- The dimensions of \(g\) can be derived as follows:
\[ [g] = \frac{[G][M]}{[L^2]} = \frac{[M^{-1}L^3T^{-2}][M]}{[L^2]} = [M^0 L^1 T^{-2}] \]
- Thus, the dimensions of acceleration due to gravity are \(M^0 L^1 T^{-2}\) (no negative mass dimension).

5. **Conclusion**:
- After analyzing all options, we find that the gravitational constant is the only quantity that has negative dimensions in mass.
- Therefore, the correct answer is **Option 2: Gravitational Constant**.