From the following physical quantities, which is the only quantity which has negative dimensions of mass?

Options

A. Linear momentum
B. Gravitationsl constant
C. Force
D. Work

Explanation

To solve the question, "From the following physical quantities, which is the only quantity which has negative dimensions of mass?" we will analyze the dimensions of each option provided.

### Step-by-Step Solution:

1. **Identify the Given Options:**
The options provided are:
- (a) Linear Momentum
- (b) Gravitational Constant
- (c) Force
- (d) Work

2. **Determine the Dimensions of Each Quantity:**

**(a) Linear Momentum:**
- Linear momentum (P) is defined as the product of mass (M) and velocity (V).
- The formula is: \( P = M \cdot V \)
- The dimension of velocity (V) is \( L T^{-1} \).
- Therefore, the dimension of linear momentum is:
\[ [P] = [M] \cdot [V] = M \cdot (L T^{-1}) = M L T^{-1} \]
- Here, the dimension of mass is positive.

**(b) Gravitational Constant:**
- The gravitational constant (G) can be derived from the formula for gravitational force:
\[ F = G \frac{M_1 M_2}{R^2} \]
- Rearranging gives:
\[ G = \frac{F \cdot R^2}{M_1 \cdot M_2} \]
- The dimension of force (F) is \( M L T^{-2} \) and the dimension of distance (R) is \( L \).
- Therefore, the dimension of G is:
\[ [G] = \frac{[F] \cdot [R^2]}{[M]^2} = \frac{(M L T^{-2}) \cdot (L^2)}{M^2} = \frac{M L^3 T^{-2}}{M^2} = M^{-1} L^3 T^{-2} \]
- Here, the dimension of mass is negative.

**(c) Force:**
- The dimension of force is already calculated as:
\[ [F] = M L T^{-2} \]
- Here, the dimension of mass is positive.

**(d) Work:**
- Work (W) is defined as the product of force and displacement:
\[ W = F \cdot d \]
- Therefore, the dimension of work is:
\[ [W] = [F] \cdot [d] = (M L T^{-2}) \cdot (L) = M L^2 T^{-2} \]
- Here, the dimension of mass is positive.

3. **Conclusion:**
- Among the options analyzed, the gravitational constant (G) is the only quantity that has negative dimensions of mass, which is \( M^{-1} L^3 T^{-2} \).

### Final Answer:
The only quantity which has negative dimensions of mass is **(b) Gravitational Constant**.