The pair of physical quantities not having the same dimensional formula is

Options

A. Acceleration, gravitational field strength
B. Torque, angular momentum
C. Pressure , modulus of elasticity
D. All the above

Answer

To determine which pair of physical quantities does not have the same dimensional formula, we will analyze each option provided in the question.

### Step 1: Analyze Option A
- **Quantities**: Acceleration and Gravitational Field Strength
- **Dimensional Formula of Acceleration**:
- Acceleration is defined as the change in velocity per unit time. Its formula is given by:
\[ \text{Acceleration} = \frac{\text{Velocity}}{\text{Time}} = \frac{\text{Displacement}}{\text{Time}^2} = \frac{L}{T^2} \]
- Therefore, the dimensional formula of acceleration is:
\[ [A] = L T^{-2} \]
- **Dimensional Formula of Gravitational Field Strength**:
- Gravitational field strength is also defined as acceleration due to gravity, hence it has the same dimensional formula:
\[ [g] = L T^{-2} \]
- **Conclusion for Option A**: Both quantities have the same dimensional formula.

### Step 2: Analyze Option B
- **Quantities**: Torque and Angular Momentum
- **Dimensional Formula of Torque**:
- Torque is defined as the product of force and the distance from the pivot point:
\[ \text{Torque} = \text{Force} \times \text{Distance} = (M L T^{-2}) \times L = M L^2 T^{-2} \]
- Therefore, the dimensional formula of torque is:
\[ [\tau] = M L^2 T^{-2} \]
- **Dimensional Formula of Angular Momentum**:
- Angular momentum is defined as the product of moment of inertia and angular velocity:
\[ \text{Angular Momentum} = \text{Moment of Inertia} \times \text{Angular Velocity} = (M L^2) \times (T^{-1}) = M L^2 T^{-1} \]
- Therefore, the dimensional formula of angular momentum is:
\[ [L] = M L^2 T^{-1} \]
- **Conclusion for Option B**: The dimensional formulas are different:
- Torque: \( M L^2 T^{-2} \)
- Angular Momentum: \( M L^2 T^{-1} \)

### Step 3: Analyze Option C
- **Quantities**: Pressure and Young's Modulus
- **Dimensional Formula of Pressure**:
- Pressure is defined as force per unit area:
\[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]
- **Dimensional Formula of Young's Modulus**:
- Young's modulus is defined as stress (force per unit area) over strain (dimensionless):
\[ \text{Young's Modulus} = \frac{\text{Stress}}{\text{Strain}} = \frac{M L^{-1} T^{-2}}{1} = M L^{-1} T^{-2} \]
- **Conclusion for Option C**: Both quantities have the same dimensional formula.

### Final Conclusion
After analyzing all options:
- Option A: Same dimensional formula.
- Option B: Different dimensional formulas (Torque and Angular Momentum).
- Option C: Same dimensional formula.

Thus, the pair of physical quantities that do not have the same dimensional formula is **Option B** (Torque and Angular Momentum).