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 pair one by one.

### Step 1: Analyze the first pair - Acceleration and Gravitational Field Strength
1. **Acceleration (a)**: The dimensional formula for acceleration is derived from the formula \( a = \frac{v}{t} \), where \( v \) is velocity (length per time).
- Dimensional formula:
\[ [a] = [L T^{-2}] = [M^0 L^1 T^{-2}] \]

2. **Gravitational Field Strength (g)**: The gravitational field strength is defined as the force per unit mass. The force has the dimensional formula \( [F] = [M L T^{-2}] \) and mass has the dimensional formula \( [M] \).
- Dimensional formula:
\[ [g] = \frac{[F]}{[M]} = \frac{[M L T^{-2}]}{[M]} = [L T^{-2}] \]
- Thus, the dimensional formula for gravitational field strength is also:
\[ [g] = [M^0 L^1 T^{-2}] \]

**Conclusion for the first pair**: Both acceleration and gravitational field strength have the same dimensional formula: \( [M^0 L^1 T^{-2}] \).

### Step 2: Analyze the second pair - Torque and Angular Momentum
1. **Torque (τ)**: Torque is defined as the product of force and the distance from the pivot point. Its dimensional formula can be derived from:
\[ [\tau] = [F] \cdot [L] = [M L T^{-2}] \cdot [L] = [M L^2 T^{-2}] \]

2. **Angular Momentum (L)**: Angular momentum is defined as the product of moment of inertia and angular velocity. Its dimensional formula is:
\[ [L] = [I] \cdot [ω] = [M L^2] \cdot [T^{-1}] = [M L^2 T^{-1}] \]

**Conclusion for the second pair**: The dimensional formula for torque is \( [M^1 L^2 T^{-2}] \) and for angular momentum is \( [M^1 L^2 T^{-1}] \). Therefore, they do not have the same dimensional formula.

### Step 3: Analyze the third pair - Pressure and Modulus of Elasticity
1. **Pressure (P)**: Pressure is defined as force per unit area. Its dimensional formula is:
\[ [P] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

2. **Modulus of Elasticity (E)**: The modulus of elasticity is also defined as stress (force per unit area) divided by strain (dimensionless). Its dimensional formula is:
\[ [E] = \frac{[P]}{[1]} = [M L^{-1} T^{-2}] \]

**Conclusion for the third pair**: Both pressure and modulus of elasticity have the same dimensional formula: \( [M^1 L^{-1} T^{-2}] \).

### Final Conclusion:
The pair of physical quantities that do not have the same dimensional formula is **Torque and Angular Momentum**.