The dimensional formula for which of the following pair is not the same ?

Options

A. impulse and momentum
B. torque and work
C. stress and pressure
D. momentum and angular momentum

Answer

To solve the question regarding which pair of physical quantities has different dimensional formulas, we will analyze each pair step by step.

### Step 1: Analyze Impulse and Momentum
1. **Impulse (J)**:
- Impulse is defined as the product of force and time.
- The formula for impulse is:
\[ J = F \cdot t \]
- The dimension of force (F) is given by:
\[ F = m \cdot a = m \cdot \left( \frac{L}{T^2} \right) = MLT^{-2} \]
- Therefore, the dimension of impulse is:
\[ [J] = [F] \cdot [t] = (MLT^{-2}) \cdot T = MLT^{-1} \]

2. **Momentum (P)**:
- Momentum is defined as the product of mass and velocity.
- The formula for momentum is:
\[ P = m \cdot v \]
- The dimension of velocity (v) is:
\[ v = \frac{L}{T} \]
- Therefore, the dimension of momentum is:
\[ [P] = M \cdot \left( \frac{L}{T} \right) = MLT^{-1} \]

**Conclusion**: The dimensions of impulse and momentum are the same: \( MLT^{-1} \).

### Step 2: Analyze Torque and Work
1. **Torque (τ)**:
- Torque is defined as the product of force and the distance from the pivot point.
- The formula for torque is:
\[ τ = F \cdot r \]
- Using the dimension of force:
\[ [τ] = [F] \cdot [r] = (MLT^{-2}) \cdot L = ML^2T^{-2} \]

2. **Work (W)**:
- Work is defined as the product of force and displacement.
- The formula for work is:
\[ W = F \cdot d \]
- Therefore, the dimension of work is:
\[ [W] = [F] \cdot [d] = (MLT^{-2}) \cdot L = ML^2T^{-2} \]

**Conclusion**: The dimensions of torque and work are the same: \( ML^2T^{-2} \).

### Step 3: Analyze Stress and Pressure
1. **Stress**:
- Stress is defined as force per unit area.
- The formula for stress is:
\[ \text{Stress} = \frac{F}{A} \]
- The dimension of area (A) is:
\[ A = L^2 \]
- Therefore, the dimension of stress is:
\[ [\text{Stress}] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = M L^{-1} T^{-2} \]

2. **Pressure**:
- Pressure is also defined as force per unit area.
- The formula for pressure is:
\[ \text{Pressure} = \frac{F}{A} \]
- Therefore, the dimension of pressure is:
\[ [\text{Pressure}] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = M L^{-1} T^{-2} \]

**Conclusion**: The dimensions of stress and pressure are the same: \( M L^{-1} T^{-2} \).

### Step 4: Analyze Momentum and Angular Momentum
1. **Momentum (P)**:
- We already calculated:
\[ [P] = MLT^{-1} \]

2. **Angular Momentum (L)**:
- Angular momentum is defined as the product of linear momentum and the distance from the pivot point.
- The formula for angular momentum is:
\[ L = P \cdot r \]
- Therefore, the dimension of angular momentum is:
\[ [L] = [P] \cdot [r] = (MLT^{-1}) \cdot L = ML^2T^{-1} \]

**Conclusion**: The dimensions of momentum and angular momentum are different: \( MLT^{-1} \) and \( ML^2T^{-1} \) respectively.

### Final Answer
The pair with different dimensional formulas is **Momentum and Angular Momentum**.